Apparatus, systems and methods including nonbinary low density parity check coding for enhanced multicarrier underwater acoustic communications

ABSTRACT

Advantageous underwater acoustic (UWA) apparatus, systems and methods are provided according to the present disclosure. The apparatus, systems and methods employ nonbinary low density parity check (LDPC) codes that achieve excellent performance and match well with the underlying modulation. The nonbinary LDPC codes of the proposed apparatus, systems and methods are formed, at least in part, from a generator matrix that has a high density to reduce the peak-to-average-power ratio (PAPR) with minimal overhead. The disclosed apparatus, systems and methods employ nonbinary regular LDPC cycle codes if the constellation is large and nonbinary irregular LDPC codes if the constellation is small or moderate. The nonbinary irregular and regular LDPC codes enable: i) parallel processing in linear-time encoding; ii) parallel processing in sequential belief propagation decoding; and iii) considerable resource reduction on the code storage for encoding and decoding.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional App. Ser. No.61/164,140 filed Mar. 27, 2009, the entire contents of which is hereinincorporated by reference in its entirety.

STATEMENT OF GOVERNMENT SUPPORT

The United States government may hold license and/or other rights inthis disclosure as a result of financial support provided bygovernmental agencies in the development of aspects of the disclosure.Parts of this work were supported by the following grants: Office ofNaval Research Grant No. N00014-07-1-0429, Office of Naval ResearchYoung Investigator Program Grant No. N00014-07-1-0805, and NationalScience Foundation Grant No. ECS-0725562.

BACKGROUND

1. Technical Field

The present disclosure relates to the field of underwater acoustic (UWA)communications. More particularly, the present disclosure relates toenhanced multicarrier UWA communications using nonbinary low densityparity check (LDPC) codes (e.g., regular or irregular LDPC codes).

2. Background Art

In general, underwater acoustic (UWA) communication (e.g., the sendingand/or receiving of acoustic signals underwater) is a difficult andcomplex process. The unique characteristics of water as a propagationmedium typically contributes to the problematic nature of UWAcommunication. For example, due to factors such as multi-pathpropagation and time variations of the channel, it is necessary toaccount for, inter alia, small available bandwidth and strong signalattenuation. Moreover, slow propagation speeds typically associated withacoustic signals may lead to significant Doppler shifts and spreading.Thus, UWA communication systems are often times limited by reverberationand time variability beyond the capability of receiver algorithms.

Multicarrier underwater acoustic communication, in the form oforthogonal frequency division multiplexing (OFDM), can be used toaddress some of the difficulties associated with UWA communications.See, e.g., M. Chitre, S. H. Ong, and J. Potter, “Performance of codedOFDM in very shallow water channels and snapping shrimp noise,” inProceedings of MTS/IEEE OCEANS, vol. 2, 2005, pp. 996-1001; P. J.Gendron, “Orthogonal frequency division multiplexing with on-offkeying:Noncoherent performance bounds, receiver design and experimentalresults,” U.S. Navy Journal of Underwater Acoustics, vol. 56, no. 2, pp.267-300, April 2006; M. Stojanovic, “Low complexity OFDM detector forunderwater channels,” in Proc. of MTS/IEEE OCEANS conference, Boston,Mass., Sep. 18-21, 2006; and B. Li, S. Zhou, M. Stojanovic, and L.Freitag, “Pilot-tone based ZPOFDM demodulation for an underwateracoustic channel,” in Proc. Of MTS/IEEE OCEANS conference, Boston,Mass., Sep. 18-21, 2006. OFDM has typically been used because of itscapability to handle high-rate transmissions over long dispersivechannels. In general, OFDM divides the available bandwidth into a largenumber of overlapping subbands, so that the symbol duration is longcompared to the multipath spread of the channel. As a result,inter-symbol-interference (ISI) may be neglected in each subband, whichreduces the complexity of channel equalization at the receiver.

Some of the research associated with OFDM UWA technologies has beenfocused on how to make OFDM work in the presence of fast channelvariations. Experimental results of researchers in the field havedemonstrated that OFDM is feasible and flexible for underwater acousticchannels. See, e.g., B. Li, S. Zhou, M. Stojanovic, L. Freitag, and P.Willett, “Multicarrier communications over underwater acoustic channelswith nonuniform Doppler shifts,” IEEE J. Oceanic Eng., vol. 33, no. 2,April 2008; B. Li, J. Huang, S. Zhou, K. Ball, M. Stojanovic, L. Freitagand P. Willett, “MIMO-OFDM for High Rate Underwater AcousticCommunications,” IEEE Journal on Oceanic Engineering, vol. 34, no. 4,pp. 634-644, October 2009; and B. Li, S. Zhou, J. Huang, and P. Willett,“Scalable OFDM design for underwater acoustic communications,” in Proc.of Intl. Conf. on ASSP, Las Vegas, Nev., Mar. 3-Apr. 4, 2008.

However, two main hurdles should be adequately addressed to successfullydeploy OFDM in a practical system: 1) Plain (or uncoded) OFDM has poorperformance in the presence of channel fading, since it typically doesnot exploit the frequency diversity inherent in the channel; and 2) OFDMtransmission typically has a high peak-to-average-power ratio (PAPR),and thus a large power backoff reduces the power efficiency and limitsthe transmission range.

Dedicated studies of coding for underwater acoustic communication arelimited. Typically, UWA communication systems employ coding schemesknown in the art. For example, trellis coded modulation (TCM) has beenused together with single carrier transmission and equalization. See,e.g., M. Stojanovic, J. A. Catipovic, and J. G. Proakis, “Phase-coherentdigital communications for underwater acoustic channels,” IEEE Journalof Oceanic Engineering, vol. 19, no. 1, pp. 100-111, January 1994.Similarly, convolutional codes and Reed Solomon (RS) codes have alsobeen examined for applications in underwater acoustic communication.See, e.g., A. Goalic, J. Trubuil, and N. Beuzelin, “Channel coding forunderwater acoustic communication system,” in Proc. of OCEANS, Boston,Mass., Sep. 18-21, 2006. Further, space time trellis codes and Turbocodes in conjunction with spatial multiplexing have been used for asingle-carrier underwater system with multiple transmitters. See, e.g.,S. Roy, T. M. Duman, V. McDonald, and J. G. Proakis, “High ratecommunication for underwater acoustic channels using multipletransmitters and space-time coding: Receiver structures and experimentalresults,” IEEE Journal of Oceanic Engineering, vol. 32, no. 3, pp.663-688, July 2007. In regards to the coding of the OFDM signal,serially concatenated convolutional codes have been used and tested witha non-iterative receiver. See, e.g., M. Chitre, S. H. Ong, and J.Potter, “Performance of coded OFDM in very shallow water channels andsnapping shrimp noise,” in Proceedings of MTS/IEEE OCEANS, vol. 2, 2005,pp. 996-1001.

Low density parity check (LDPC) codes are known to be capacity-achievingcodes. See, e.g., R. G. Gallager, Low Density Parity Check Codes.Cambridge, Mass.: MIT Press, 1963. LDPC codes have been extensivelystudied for wireless radio systems. Relative to binary LDPC codes, oneadvantage of nonbinary LDPC codes is that they can be matched very wellwith underlying modulation. For example, nonbinary LDPC codes were firstcombined with high order modulation in radio communication systems withtwo transmitters and two receivers. See. e.g., F. Guo and L. Hanzo, “Lowcomplexity non-binary LDPC and modulation schemes communicating overMIMO channels,” in Proc. of VTC, vol. 2, pp. 1294-1298, Sep. 26-29,2004. Further, simulations have shown that an iterative receiver withnonbinary LDPC codes over GF(16) can outperform the best optimizedbinary LDPC code in both performance and complexity, while anon-iterative receiver with regular LDPC cycle code over GF(256) canachieve much better performance with comparable decoding complexitycompared to the binary iterative system. See, e.g., R.-H. Peng and R.-R.Chen, “Design of nonbinary LDPC codes over GF(q) for multiple-antennatransmission,” in Proc. of Military Communications conference 2006,Washington, D.C., Oct. 23-25 2006, pp. 1-7.

Current OFDM UWA communication systems fail to adequately address theshortcomings of OFDM technologies. Specifically, uncoded or plain OFDMhas poor performance in the presence of channel fading and OFDMtransmission has a high peak-to-average-power ratio (PAPR). Due to thelimited bandwidth, high order constellations are more desirable formulticarrier underwater communication. These and other inefficienciesand opportunities for improvement are addressed and/or overcome by theapparatus, systems and methods (e.g., LDPC based apparatus, systems andmethods) of the present disclosure.

SUMMARY

The present disclosure relates to apparatus, systems and methods forfacilitating enhanced underwater acoustic (UWA) communications. Moreparticularly, the present disclosure involves apparatus, systems andmethods for UWA communications that utilize, at least in part, nonbinarylow density parity check (LDPC) codes. In some embodiments, thenonbinary low density parity check codes are irregular, while in otherembodiments the nonbinary low density parity check codes are regular.The disclosed approaches use irregular and/or regular nonbinary LDPCcodes to address at least two main issues in underwater acoustic OFDMcommunication: (i) plain OFDM has poor performance in the presence ofchannel fading; and iii) OFDM transmission has a highpeak-to-average-power ratio (PAPR). Some embodiments of the presentdisclosure include LDPC codes formed from a generator matrix that has ahigh density, and thus reduces the PAPR considerably with minimaloverhead.

In some embodiments, nonbinary irregular LDPC codes are employed, forinstance with small or moderate sized constellations (e.g., BPSK, QPSK,8-QAM and 16-QAM and/or Galois Fields GF(q) where q<64). In oneembodiment, a large portion of the parity check matrix of the irregularLDPC codes resembles that of regular LDPC cycle codes, thereby retainingmany of the benefits of regular LDPC cycle codes. The other portion ofthe parity check matrix of the irregular LDPC codes includes a columnweight greater than that of the parity check matrix of the regular LDPCcycle codes (i.e., a column weight of greater than 2). Therefore, theirregular LDPC cycle codes can be formed by replacing a portion of theparity check matrix of the regular LDPC codes H with columns of a weightgreater than 2. In this way, the irregular LDPC codes can be arranged ina split representation, wherein H₁ contains all weight-2 columns and H₂contains all of the columns of a weight greater than 2, therebyimproving performance while retaining at least some of the benefits ofregular LDPC cycle codes. Of note, simulation and experimental resultsconfirm the excellent performance of the proposed nonbinary irregularLDPC codes. Advantageous design of irregular LDPC codes is alsodisclosed.

In other embodiments, regular LDPC cycle codes are employed, forinstance with large sized constellations (e.g., 64-QAM and/or GaloisFields GF(q) where q≧64). The regular LDPC cycle codes may be employedover GF(q), whose parity check matrix H has fixed column weight j=2 andfixed row weight d. Therefore, the term “nonbinary regular LDPC cyclecodes” is used herein to refer to nonbinary LDPC codes that are “cyclecodes” in the sense that they have a parity check matrix with a columnweight of 2 and “regular” in the sense that they are further constrainedwith equal weight on all rows.

In this embodiment, any regular cycle GF(q) code's parity check matrix Hcan be put into a concatenation form of row-permuted block-diagonalmatrices after row and column permutations if d is even, or, if d is oddand the code's associated graph contains at least one spanning subgraphthat consists of disjoint edges. The equivalent representation of H mayenable: i) parallel processing in linear-time encoding; ii) parallelprocessing in sequential belief propagation decoding, which increasesthe throughput without compromising performance or complexity; and iii)considerable resource reduction on the code storage for encoding anddecoding.

Advantageous design of regular cycle GF(q) codes—that achieve excellentperformance, match well with the underlying modulation, and can beencoded in linear time and in parallel—are also disclosed. In oneembodiment, the design of regular cycle GF(q) codes consists of thestructure design of H and selection of nonzero entries. Three differentmethodologies may be used to determine the design of the regular cycleGF(q) codes: i) design based on known graphs; ii) computer search basedalgorithms; and iii) interleaver design based on the equivalentrepresentation of H. In some embodiments, the selection of nonzeroentries effectively lowers the performance error floor.

Additional features, functions and benefits of the disclosed apparatus,systems and methods will be apparent from the description which follows,particularly when read in conjunction with the appended figures.

BRIEF DESCRIPTION OF THE DRAWINGS

To assist those of ordinary skill in the art in making and using thedisclosed apparatus, systems and methods, reference is made to theappended figures, wherein:

FIG. 1 illustrates a schematic block diagram of a nonbinary low densityparity check (LDPC) coded OFDM system.

FIG. 2 a depicts an exemplary check matrix over GF(8) with column weightj=2 and row height d=4.

FIG. 2 b depicts the associated graph of the exemplary check matrix ofFIG. 2 a.

FIG. 3 depicts a 2-factor graph of the associated graph of FIG. 2 b.

FIG. 4 a depicts a 1-factor split graph from the 2-factor graph of FIG.3.

FIG. 4 b depicts the companion 1-factor split graph of FIG. 4 a from the2-factor graph of FIG. 3.

FIG. 5 illustrates a performance comparison of exemplary nonbinaryirregular codes over GF(16) and mean column weights.

FIG. 6 illustrates a performance comparison of exemplary nonbinaryirregular codes over GF(16) and exemplary binary optimized LDPC codes.

FIG. 7 a depicts an exemplary uneven 2-factor graph which contains onelength-4 cycle C₁=v₁e₁v₂e₂v₃e₃v₄e₄v₁ and one length-5 cycleC₂=v₅e₅v₆e₆v₇e₇v₈e₈v₉e₉v₅.

FIG. 7 b depicts the 2-factor graph of FIG. 7 a partitioned into threeorthogonal groups {e₁,e₃,e₅,e₇}, {e₂,e₄,e₆,e₈} and {e₉}.

FIG. 8 depicts a performance comparison of exemplary regular, irregularand bipartite regular cycle GF(q) codes under standard beliefpropagation (BP) decoding up to 80 iterations where the code rate is ½and the codeword length is 1008 bits.

FIG. 9 depicts a performance comparison of exemplary sequential andstandard BP decodings for the regular and bipartite regular cycle codesshown in FIG. 8.

FIG. 10 depicts a performance comparison on the average number ofiterations of exemplary sequential BP decoding and standard BP decodingfor the exemplary regular and bipartite regular cycle codes shown inFIG. 8.

FIG. 11 depicts a performance comparison of exemplary cycle codes withdifferent selections on nonzero entries under standard BP decoding up to80 iterations with a codeword length of 1008 bits.

FIG. 12 depicts a performance comparison of exemplary regular cyclecodes using semi-random interleavers and the progressive edge-growth(PEG) method with a codeword length of 1344 bits.

FIG. 13 a depicts the block error rate (BLER) performance of exemplaryLDPC codes of different modes over an AWGN channel.

FIG. 13 b depicts the bit error rate (BER) performance of exemplary LDPCcodes of different modes over an AWGN channel.

FIG. 14 depicts the BLER and BER performance of all the modes over OFDMRayleigh fading channel and the uncoded BER curves for differentmodulations of exemplary nonbinary LPDC codes.

FIG. 15 depicts the BLER and BER performance of all the modes over OFDMRayleigh fading channel and the uncoded BER curves for differentmodulations of exemplary nonbinary LPDC codes.

FIG. 16 depicts a comparison of exemplary LDPC and CC codes of rate 1/2under different modulation over an OFDM Rayleigh fading channel.

FIG. 17 depicts a comparison of PAPR reduction using exemplary LDPC andconvolutional codes (“CC”).

FIG. 18 depicts another comparison of PAPR reduction using exemplaryLDPC and CC codes using a rate of ½ coding.

FIG. 19 depicts a performance comparison of exemplary LDPC codes ofdifferent coded modulation schemes over an AWGN channel.

FIG. 20 depicts a performance comparison of exemplary LDPC codes ofdifferent coded modulation schemes over a Rayleigh fading channel.

FIG. 21 depicts a comparison of exemplary LDPC and CC codes of rate ½coding under different modulation over an AWGN channel.

FIG. 22 depicts coded BER with 16-QAM constellation and rate of ½ codingof exemplary LDPC codes.

FIG. 23 depicts coded BER as a function of a number of receive-elementsaveraged over data collected from 13 days in an experiment of exemplaryLDPC codes.

FIG. 24 depicts BLER as a function of a number of receive-elementsaveraged over data collected from 13 days in an experiment of exemplaryLDPC codes.

FIG. 25 depicts bit error rates in different Julian dates, North 1000 m,8 receiver-elements and 16-QAM of exemplary LDPC codes.

FIG. 26 depicts bit error rates in different Julian dates, North 1000 m,8 receiver-elements and 64-QAM of exemplary LDPC codes.

DESCRIPTION OF EXEMPLARY EMBODIMENT(S)

The present disclosure provides for advantageous apparatus, systems andmethods for facilitating enhanced underwater acoustic (UWA)communications. More particularly, the disclosed apparatus, systems andmethods generally involve nonbinary irregular and regular low densityparity check (LDPC) codes. Advantageously, irregular LDPC cycle codesare employed with small or moderate sized constellations (e.g., BPSK,QPSK, 8-QAM and 16-QAM and/or Galois Fields GF(q) where q is less thanabout 64) and regular LDPC codes are employed with large sizedconstellations (e.g., 64-QAM and/or Galois Fields GF(q) where q isgreater than or equal to about 64). In general, the regular LDPC codeshave a parity check matrix that has a fixed column width weight 2 and afixed row weight d (hereinafter referred to as “cycle” codes). In anexemplary embodiment, the parity check matrix of the regular cycle codecan be placed into a concatenation form of row-permuted block diagonalmatrices after row and column permutations if d is even, or, if d is oddand the code's associated graph contains at least one spanning subgraphthat consists of disjoint edges.

In another embodiment, a large portion of the parity check matrix of theirregular LDPC codes resembles that of regular LDPC cycle codes, therebyretaining many of the benefits of regular LDPC codes. The remainingportion of the parity check matrix of the irregular LDPC codes includesa column weight greater than that of the parity check matrix of theregular LDPC codes (e.g., a column weight of greater than 2). Therefore,the irregular LDPC codes can be formed by replacing a portion of theparity check matrix of the regular LDPC codes cycle with columns of aweight greater than 2. In this way, the irregular LDPC codes can bearranged in a split representation—e.g., a matrix with weight-2 columnsand a matrix wherein the columns are of a weight greater than 2. In thismanner the irregular LDPC codes improve performance while retaining atleast some of the benefits of regular LDPC codes.

The embodiments of the disclosed apparatus, systems and methods employthe nonbinary regular and irregular LDPC codes to enable parallelprocessing in linear-time encoding and parallel processing in sequentialbelief propagation decoding, which increases the throughput withoutcompromising performance or complexity. Embodiments of the LDPC codesachieve excellent performance, match well with the underlying modulationand/or reduce the PAPR considerably with minimal overhead. Oneembodiment of the disclosed PAPR reduction approach requires multiplerounds of encoding for each information block at the transmitter, hence,the fast and parallel encoding algorithm for the proposed nonbinary LDPCcodes is well suited. All publications, applications, patents, figuresand other references mentioned herein are incorporated by reference intheir entirety.

1. The System, Method and Apparatus

FIG. 1 shows the block diagram of an exemplary underwater OFDM systemwith nonbinary LDPC coding. Encoding and decoding are performed for eachOFDM block separately. See, e.g., B. Li, S. Zhou, M. Stojanovic, L.Freitag, and P. Willett, “Multicarrier communications over underwateracoustic channels with nonuniform Doppler shifts,” IEEE J. Oceanic Eng.,vol. 33, no. 2, April 2008. In theory, if an LDPC code over GF(q) isused where q=2^(p), then {α₀=0, α₁, . . . , α_(q-1)} denotes elements inGF(q). Also, a constellation size of M=2^(b) may be used by the OFDMmodulator. One advantage of nonbinary LDPC coding is that the fieldorder can be matched with the constellation size, i.e., p=b. In thismanner, one element in GF(q) can be mapped to one point in the signalconstellation. In an embodiment where b is small, it may be preferableto choose p>b. Further, if it is assumed that J:=p/b is an integer, eachelement in GF(q) will be mapped to J symbols drawn from theconstellation. Therefore, the mapper may be described as:α_(i)→[φ⁰(α_(i)), . . . , φ^(J-1)(α_(i))], i=0, . . . , q−1  (1)where φ^(j)(α_(i)) is one point in the signal constellation. It can alsobe assumed that K_(d) subcarriers are used for data transmission, andthe LDPC code rate is r.

Applying the above mentioned assumptions, the transmitter can be said tooperate as follows. First, for each OFDM block, rbK_(d) information bitsare mapped to rbK_(d)/p symbols in GF(q), with every p bits mapped to asingle GF(q) symbol through a bit-to-symbol mapper g. Then, the LDPCencoder outputs bK_(d)/p coded symbols in GF(q), which pass through acoded-symbol interleaver π to obtain a vectoru=[u[0], . . . , u[K _(d) /J−1]]^(T).  (2)In this way, the mapper in the expression enumerated as (1) above, isable to map the vector u to a modulated-symbol vector s:=[s[0] . . . ,s[Kd−1]]^(T) as:s=[φ ⁰(u[0]), . . . , φ^(J-1)(u[0]),φ⁰(u[1]), . . . , φ^(J-1)(u[K _(d)/J−1])]^(T).  (3)The Kd entries of s are thus distributed to the OFDM data subcarriers.An OFDM transmission is then formed after mixing the data subcarrierswith pilot and null subcarriers. See, e.g., B. Li, S. Zhou, M.Stojanovic, L. Freitag, and P. Willett, “Multicarrier communicationsover underwater acoustic channels with nonuniform Doppler shifts,” IEEEJ. Oceanic Eng., vol. 33, no. 2, April 2008, which is hereby expresslyincorporated by reference in its entirety. Using a block-by-block OFDMreceiver (such as the one described in the publication cited above) theequivalent channel input-output model on the data subcarriers may beexpressed as:y[k]=H[k]s[k]+n[k], k=0, . . . , K _(d)−1,  (4)where H[k] is the channel frequency response on the kth data subcarrier,y[k] is the output on the kth data subcarrier, and n[k] is the compositenoise with contributions from ambient noise, the residual inter-carrierinterference (ICI), and the noise induced by channel estimation error.In theory, it can be assumed that n[k] has variance σ² per real andimaginary dimension. Thus, the average signal to noise ratio can bedefined as

$\begin{matrix}{{\left| {E_{s}/N_{0}} \right. = \frac{{E_{m} \cdot E}\left\{ {{\hat{H}\lbrack k\rbrack}}^{2} \right\}}{2\sigma^{2}}},} & (5)\end{matrix}$where E_(m) is the average symbol energy of the constellation, and |.|denotes the absolute value of a complex number, and E{.} denotes theexpectation operation.

When the noise variance σ² is available, the demapper can compute thelikelihood

$\begin{matrix}{{{\Pr\left( {{u\lbrack k\rbrack} = \alpha_{i}} \right)} \propto {\exp\left( \frac{- {\sum\limits_{j = 0}^{J - 1}{{{y\left\lbrack {{k\; J} + j} \right\rbrack} - {{H\left\lbrack {{k\; J} + j} \right\rbrack}{\phi^{j}\left( \alpha_{i} \right)}}}}^{2}}}{2\sigma^{2}} \right)}},\mspace{14mu}{k = 0},\ldots\mspace{14mu},{{{K_{d}/J} - 1};\mspace{14mu}{1 = 0}},\ldots\mspace{14mu},{q - 1.}} & (6)\end{matrix}$The likelihood values can then be passed to the deinterleaverπ⁻⁻¹/before being passed to the LDPC decoder. The FFT-based q-arysum-product algorithm (FFT-QSPA) may be used for iterative decoding.See, e.g., H. Song and J. R. Cruz, “Reduced-complexity decoding of q-aryLDPC codes for magnetic recording,” IEEE Trans. Magn., vol. 39, pp.1081-1087, 2003. In an exemplary embodiment, after a finite number ofdecoding iterations, hard decisions on the nonbinary symbols are made atthe output of the LDPC decoder, based on which information bits arefound. Unlike a system with binary coding and high order modulation, theproposed system in FIG. 1 and described herein does not require anyiterative processing between the demapper and the LDPC decoder.

When the noise variance is not available, the demapper can compute thelog-likelihood-ratio vector (LLRV) over GF(q). The LLRV of u[k] isdefined as z[k]=[z₀[k], z₁[k], . . . , z_(q-1)[k]]^(T), where

$\begin{matrix}{{z_{i}\lbrack k\rbrack} = {\ln{\frac{\Pr\left( {{u\lbrack k\rbrack} = \alpha_{i}} \right)}{\Pr\left( {{u\lbrack k\rbrack} = 0} \right)}.}}} & (7)\end{matrix}$

From equation (6), it can be determined that

$\begin{matrix}{{z_{i}\lbrack k\rbrack} = {{- \frac{1}{2\sigma^{2}}}{\sum\limits_{j = 0}^{J - 1}{\left( {{{{y\left\lbrack {{k\; J} + j} \right\rbrack} - {{\hat{H}\left\lbrack {{k\; J} + j} \right\rbrack}{\phi^{j}\left( \alpha_{i} \right)}}}}^{2} - {{{y\left\lbrack {{k\; J} + j} \right\rbrack} - {{\hat{H}\left\lbrack {{k\; J} + j} \right\rbrack}{\phi^{j}(0)}}}}^{2}} \right).}}}} & (8)\end{matrix}$

In an exemplary embodiment, the LLRV values are passed to thedeinterleaver π⁻¹ before being passed to the LDPC decoder. The min-sum(MS), or extended min-sum (EMS) algorithms can be used for iterativedecoding. See, e.g., D. Declercq and M. Fossorier, “Decoding algorithmsfor nonbinary LDPC codes over GF(q),” IEEE Trans. Commun., vol. 55, no.4, pp. 633-643, April 2007; and A. Voicila, D. Declercq, F. Verdier, M.Fossorier, and P. Urard, “Low complexity, low-memory EMS algorithm fornon-binary LDPC codes,” in Proc. IEEE International Conf. on Commun.,Glasgow, Scotland, Jun. 24-28 2007, pp. 671-676. It is noted that theLLRV generated by the expression enumerated as (8) above is proportionalto the reciprocal of σ², and the updating rules of the MS (or EMS)decoding algorithm at the check nodes and variable nodes are linearoperations with respect to the reciprocal of σ². Therefore, all themessages exchanged during decoding iterations can be proportional to thereciprocal of σ² and the decoding results may remain unchanged with σ²set to an arbitrary value.

It is also noted that when the code alphabet is matched to themodulation alphabet, i.e., p=b, or when p is an integer multiple of b,the interleaver in FIG. 1 is not necessary, as interleaving the codedsymbols amounts to shuffling the columns of the parity check matrix ofthe LDPC code; hence interleaving can be absorbed into the code design.In such cases, the proposed system in FIG. 1 does not require anyiterative processing between the demapper and the LDPC decoderregardless of the constellation labelling rules—because the demapperproduces the likelihood probabilities (or LLRV) for each coded symbolover GF(q) that are independent of other coded symbols. For otherchoices of p and b, interleaving and iterative demapping may be useful.It is further noted that for a binary LDPC coded system with high ordermodulation, (i) other constellation labelling rules (e.g., setpartitioning) can improve the system performance relative to Graylabelling, but require iterative processing between the maximum aposterior (MAP) demapper and the LDPC decoder, and (ii) the noisevariance must be estimated for demapping.

2. The Proposed Nonbinary LDPC Codes

A. Nonbinary Regular Cycle Code

Gallager's binary LDPC codes are excellent error-correcting codes thatachieve performance close to the benchmark predicted by the Shannontheory. See, e.g., R. G. Gallager, Low Density Parity Check Codes,Cambridge, Mass.: MIT Press, 1963, and D. J. C. Mackay, “Gooderror-correcting codes based on very sparse matrices,” IEEE Trans.Inform. Theory, vol. 45, no. 2, pp. 399-431, March 1999. The extensionof LDPC to non-binary Galois field GF(q) was first investigatedempirically by Davey and Mackay over the binary-input AWGN channel. See,e.g., M. C. Davey and D. Mackay, “Low-density parity-check codes overGF(q),” IEEE Commun. Lett., vol. 2, pp. 165-167, June 1999. Since then,nonbinary LDPC codes have been actively studied.

The simplest LDPC codes are cycle codes, as their parity check matriceshave column weight j=2. See, e.g., D. Jungnickel and S. A. Vanstone,“Graphical codes revisited,” IEEE Trans. Inform. Theory, vol. 43, pp.136-146, January 1997. It has been found that the mean column weight ofnonbinary LDPC codes must approach 2 when the field order qincreases—that is, the best nonbinary LDPC codes for very large q tendto be cycle codes over GF(q). See, e.g., M. C. Davey and D. Mackay,“Monte Carlo simulations of infinite low density parity check codes overGF(q),” in Proc. of Int. Workshop on Optimal Codes and related Topics,Bulgaria, Jun. 9-15 1998. Available athttp://www.inference.phy.cam.ac.uk/is/papers/; and M. C. Davey,Error-Correction using Low-Density Parity-Check Codes, Dissertation,University of Cambridge, 1999. It is also known that cycle GF(q) codescan achieve near-Shannon-limit performance as q increases and canoutperform other LDPC codes, including degree-distribution optimizedbinary irregular LDPC codes. X.-Y. Hu and E. Eleftheriou, “Binaryrepresentation of cycle tannergraph GF(2b) codes,” Proc. InternationalConference on Communications, vol. 27, no. 1, pp. 528-532, June 2004.

One main concern of nonbinary LDPC codes with large q is the decodingcomplexity. An FFT-based q-ary sum-product algorithm (FFT-QSPA) fordecoding a general LDPC code over binary extension fields has beenproposed, whose decoding complexity increases on the order of O(q logq). See. e.g., H. Song and J. R. Cruz, “Reduced-complexity decoding ofq-ary ldpc codes for magnetic recording,” IEEE Trans. Magn., vol. 39,pp. 1081-1087, March 2003; and L. Barnault and D. Declercq, “Fastdecoding algorithm for LDPC codes over GF(2^(q)),” in Proc. IEEE Inform.Theory Workshop, 2003, pp. 70-73. There also exists a min-sum versionalgorithm which works in the log-domain for nonbinary LDPC codes,similar to the min-sum decoding for binary LDPC codes where the Jaccobioperation max* is replaced by the max operation. See, e.g., H.Wymeersch, H. Steendam, and M. Moeneclaey, “Log-domain decoding of LDPCcodes over GF(q),” in Proc. IEEE Int. Conf. Commun., Paris, France, June2004, pp. 772-776. Reduced-complexity decoding algorithms for nonbinaryLDPC codes have also been recently developed. See, e.g., M. Tjader, M.Grimnell, D. Danev, and H. M. Tullberg, “Efficient message-passingdecoding of LDPC codes using vector-based messages,” in Proc.International Symp. on Inform. Theory, Seattle, Wash., July 2006, pp.1713-1717; D. Declercq and M. Fossorier, “Decoding algorithms fornonbinary LDPC codes over GF(q),” IEEE Trans. Commun., vol. 55, no. 4,pp. 633-643, April 2007; and A. Voicila, D. Declercq, F. Verdier, M.Fossorier, and P. Urard, “Low complexity, low-memory EMS algorithm fornon-binary LDPC codes,” in Proc. IEEE International Conf. on Commun.,Glasgow, Scotland, Jun. 24-28 2007, pp. 671-676. Using a geometricalvector representation and the table lookup, an efficient message-passingdecoding algorithm for nonbinary LDPC codes over M-ary phase shiftkeying (PSK) has been developed, which can perform close to the beliefpropagation decoding algorithm with far less decoding complexity.Truncating the size of extrinsic messages from q to n_(m), the extendedmin-sum (EMS) algorithm may reduce the total decoding complexity fromthe order of O(q log q) to O(n_(m) log n_(m)), where n_(m) could be muchsmaller than q. The improved version of the EMS algorithm can furtherreduce the message storage requirement.

One unique advantage of nonbinary LDPC codes over binary LDPC codes isthat nonbinary codes can match very well the underlying modulation, andbypass the need for a symbol-to-bit conversion at the receiver. Thepresent disclosure provides for apparatus, systems and methods that makeuse of LDPC codes with column weight j=2 in their parity check matrix H,termed as cycle codes. See, e.g., D. Jungnickel and S. A. Vanstone,“Graphical codes revisited,” IEEE Trans. Inform. Theory, vol. 43, pp.136-146, January 1997. Although the distance properties of binary cyclecodes are not as good as the LDPC codes of column weight j≧3, it hasbeen shown in that cycle GF(q) codes can achieve near-Shannon-limitperformance as q increases. See, e.g., R. G. Gallager, Low DensityParity Check Codes, Cambridge, Mass.: MIT Press, 1963, and X.-Y. Hu andE. Eleftheriou, “Binary representation of cycle Tanner-graph GF(2b)codes,” IEEE International Conference on Communications, vol. 27, no. 1,pp. 528-532, June 2004. Further, X.-Y. Hu et al. demonstrated numericalresults that show cycle GF(q) codes can outperform other LDPC codes,including degree-distribution-optimized binary irregular LDPC codes. Forhigh order fields (q≧64), the best GF(q)-LDPC codes decoded by beliefpropagation (BP) are commonly theorized to be ultra sparse, with a goodexample being the cycle codes that have j=2. See, e.g., M. C. Davey andD. Mackay, “Low-density parity-check codes over GF(q),” IEEE Commun.Lett., vol. 2, pp. 165-167, June 1999, and M. C. Davey, Error-Correctionusing Low-Density Parity-Check Codes, Dissertation, University ofCambridge, 1999.

Reduced complexity algorithms for decoding a general LDPC code overGF(q) have also been proposed. See, e.g., H. Song and J. R. Cruz,“Reduced-complexity decoding of Q-ary LDPC codes for magneticrecording,” IEEE Trans. Magn., vol. 39, pp. 1081-1087, March 2003, andL. Barnault and D. Declercq, “Fast decoding algorithm for LDPC codesover GF(2q),” in Proc. IEEE Inform. Theory Workshop, pp. 70-73, 2003. Auniversal linear-complexity encoding algorithm for any cycle GF(q) codehas also been determined. See, e.g., J. Huang and J.-K. Zhu, “Lineartime encoding of cycle GF(2^(p)) codes through graph analysis,” IEEECommun. Lett., vol. 10, pp. 369-371, May 2006. As such, the performanceand implementation advantages of cycle GF(q) codes make them promisingfor practical applications.

One popular representation of LDPC codes is based on the Tanner-graph,which is a bipartite graph with m constraint (check) nodes and nvariable nodes connected by edges specified by the nonzero entries inthe parity check matrix H of size m×n. See, e.g., R. M. Tanner, “Arecursive approach to low complexity codes,” IEEE Trans. Inform. Theory,vol. 27, pp. 533-547, September 1981. In preferred embodiments of theapparatus, systems and methods disclosed herein, cycle GF(q) codes canbe represented using an associated graph G with m vertices and n edges,where each vertex represents one constraint node corresponding to onerow of H, and each edge represents one variable node corresponding toone column of H. See, e.g., J. Huang and J.-K. Zhu, “Linear timeencoding of cycle GF(2p) codes through graph analysis,” IEEE Commun.Lett., vol. 10, pp. 369-371, May 2006. If the row weight of H for acycle code is fixed as d, then each vertex of its associated graph G maybe exactly connected to d edges. Such a graph is d-regular, and such aLDPC code is defined as a regular cycle code over GF(q) herein. See,e.g., D. Reinhard, Graph Theory, 2nd edition, Springer-Verlag, 2000.

In preferred embodiments, UWA communication apparatus, systems andmethods include a cycle GF(q) code—an LDPC code whose m×n parity checkmatrix H has weight j=2 for each column. As such, in the preferredembodiments the cycle GF(q) code can be represented by an associatedgraph G=(V,E) with m vertices V={v1, . . . , v_(m)} and n edges E={e₁, .. . , e_(n)}, where each vertex represents a constraint nodecorresponding to a row of H, and each edge represents a variable nodecorresponding to a column of H, as shown in FIGS. 2 a and 2 b. If thecycle GF(q) code also has a fixed row weight d in H, the graph G isd-regular in that each vertex is exactly linked to d edges. This codewill be referred to as regular cycle GF(q) code hereinafter. Of note,2n=dm for regular cycle GF(q) codes. Further, when H is full row-rank, Hdefines a regular cycle GF(q) code of rate R=(d−2)/d.

It is herein proposed that the graph theory is an advantageous way toanalyze regular cycle GF(q) codes. Before analysis, it is noted that theterm “k-factor” is defined as a k-regular spanning subgraph of G thatcontains all the vertices, and the term “k-factorable” is defined as agraph G with edge-disjoint k-factors G₁, G₂ . . . , G_(L) such thatG=G₁∪G₂ . . . , ∪G_(L). Thus, a 1-factor is a spanning subgraph thatconsists of disjoint edges, while a 2-factor is a spanning subgraph thatconsists of disjoint cycles, as shown in FIGS. 3-4 b. For a subgraph G′of G, it can be assumed that H_(G′) be the sub-matrix of H restricted tothe rows and columns indexed by the vertices and edges of G′respectively, which can be obtained from H by deleting the rows andcolumns other than those corresponding to the vertices and edges of G′respectively. Herein, H_(G′) is referred to as the sub-matrix of Hassociated with G′. In some embodiments, two sub-matrices of H areassociated with an edge and a cycle of the graph G. For each edge, thesub-matrix may be represented as:

$\begin{matrix}{{{\overset{\sim}{h}}^{e} = \begin{bmatrix}\alpha \\\beta\end{bmatrix}},} & (9)\end{matrix}$where α and β correspond to those two nonzero entries of the column of Hindexed by this edge. For a length-k cycle C that consists of kconsecutive edges e₁, e₂, . . . , e_(k), a k×k matrix may be defined as:

$\begin{matrix}{{{\overset{\sim}{H}}^{c} = \begin{bmatrix}\alpha_{1} & 0 & 0 & \ldots & \beta_{k} \\\beta_{1} & \alpha_{2} & 0 & \ldots & 0 \\0 & \beta_{2} & \alpha_{3} & \ldots & 0 \\\vdots & \vdots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & \beta_{k - 1} & \alpha_{k}\end{bmatrix}},} & (10)\end{matrix}$where α_(i)s and β_(i)s correspond to those two nonzero entries of thecolumn of H indexed by edge e_(i). For two matrices H₁ and H₂, if H₁ canbe transformed into H₂ simply through row and column permutations, thenH₁ may be deemed equivalent to H₂ and the relationship denoted as H₁≅H₂.Theorem 1

Considering the foregoing, a first theorem may be expressed as:

For a cycle GF(q) code, if its associated graph G is d-regular withd=2r, its parity check matrix H of size m×n has the equivalent formH≅[ H ₁ , P ₂ H ₂ , . . . , P _(r) H _(r)],  (11)where P_(i) is m×m permutation matrix, and H _(i) is of size m×m, 1≦i≦r.The matrix H _(i) has an equivalent block-diagonal formH _(i)≅diag({tilde over (H)} _(i,1) ^(c) , {tilde over (H)} _(i,2) ^(c), . . . , {tilde over (H)} _(i,L) _(i) ^(c)),  (12)where the matrix {tilde over (H)}_(i,1) ^(c) has the form of theexpression enumerated as (10) above and is of size k_(i,l)×k_(i,l) thatsatisfies

$m = {\sum\limits_{l = 1}^{L_{i}}\;{k_{i,l}.}}$Proof of Theorem 1

A proof of the first theorem is as follows. If G is d-regular with d=2r,r>0, G is 2-factorable. See, e.g., D. Reinhard, Graph Theory, 2ndedition, Springer-Verlag, 2000. The r edge-disjoint 2-factors of G canbe denoted by G₁, G₂, . . . , G_(r). The columns of H can be arranged insuch a pattern that the columns indexed by the edges of G₁ are placed inthe first m columns, followed by the m columns indexed by the edges ofG₂ until the m columns which are indexed by the edges of G_(r). In thisway, H is partitioned to r sub-matrices of size m×m each, arranged asH≅[H_(G1), . . . , H_(Gr)], where H_(Gi) is the sub-matrix of Hassociated with G_(i).

It can also be shown that each m×m sub-matrix H_(Gi) has an equivalentblock diagonal form as in the expression enumerated as (12) above. Each2-factor G_(i) can be decomposed into a set of disjoint cycles. It canbe assumed that G_(i) consists of L_(i) disjoint cycles C_(i,l),1≦l≦L_(i), where C_(i,l) is of length k_(i,l) that satisfies

$m = {\sum\limits_{l = 1}^{L_{i}}\;{k_{i,l}.}}$The rows and columns of H_(Gi) can be arranged in sequence of rows andcolumns indexed by C_(i,1), C_(i,2), . . . , C_(i,Li), where theresultant matrix will have a block-diagonal form diag({tilde over(H)}_(i,1) ^(c), {tilde over (H)}_(i,2), . . . , {tilde over (H)}_(i,Li)^(c)), where {tilde over (H)}_(i,l) ^(c) represents the matrixassociated with C_(i,l) and has a form as in the expression enumeratedas (11) above. Thus, it can be said that H_(Gi)=P_(i) H _(i)R_(i), whereH _(i) is defined in the expression enumerated as (12) above, and P_(i)and R_(i) are permutation matrices, 1≦i≦r.

Therefore, the matrix H can be arranged to have an equivalent form [P₁ H₁R₁,P₂ H ₂R₂, . . . , P_(r) H _(r)R_(r)], and further permute the rowsof H to let P₁ be the identity matrix and permute the columns of H_(Gi)to let each R_(i) be the identity matrix. Thus, the resultant matrixwould have a form like the expression enumerated as (11) above. Thiscompletes the proof.

Theorem 2

Considering the foregoing, a second theorem may be expressed as:

Consider a regular cycle GF(q) code with d=2r+1. If its associated graphG contains at least one 1-factor, then its parity check matrix H of sizem×n has the equivalent formH≅[ H ₁ , P ₂ H ₂ , . . . , P _(r) H _(r) , P ^(e) H ^(e)]  (13a)where P_(i)s and P^(e) are permutation matrices, H ^(e) is an m×mblock-diagonal matrix having the form as in the expression enumerated as(12) above, i=1, . . . , r, H ^(e) is an m×m/2 matrix having anequivalent block-diagonal form as

$\begin{matrix}{{{\overset{\_}{H}}^{e} \cong {{diag}\left( {{\overset{\sim}{h}}_{1}^{e},{\overset{\sim}{h}}_{2}^{e},\ldots\mspace{14mu},{\overset{\sim}{h}}_{\frac{m}{2}}^{e}} \right)}},} & \left( {13b} \right)\end{matrix}$where {tilde over (h)}_(i) ^(e) is a vector having the form as in theexpression enumerated as (9) above.Proof of Theorem 2

A proof of the second theorem is as follows. If G is d-regular withd=2r+1, r>0 and G has a 1-factor M, G′ can denote the graph obtainedfrom G by deleting the edges in M. Thus, G′ is 2r-regular. The columnsof H can be arranged in such a pattern that the columns indexed by theedges of G′ are placed in the first rm columns, followed by the m/2columns which are indexed by the edges of M. Therefore, arranged H canbe expressed as H≅[H_(G′), H_(M)], where H_(G′) is the sub-matrix of Hassociated with G′ and H_(M) is the sub-matrix of H associated with M.

Applying Theorem 1, the sub-matrix H_(G′) has a form as shown in theexpression enumerated as (11) above. The form of sub-matrix H_(M) canthen be shown. Since M is a 1-factor of G, M is a union of disjointedges. The edges of M may then be denoted by E_(i), 1≦i≦m/2. The rowsand columns of H_(M) can be arranged in sequence of rows and columnsindexed by E₁,E₂, . . . , E_(m/2), and the resultant matrix will havethe form as shown in the expression enumerated as (13b) above. Thus,H_(M)=P^(e) H ^(e)R^(e), where H ^(e) is defined in the expressionenumerated as (13b) above, and P^(e) and R^(e) are permutation matrices.

Therefore, the matrix H would have an equivalent form like └H₁,P₂H₂, . .. , P_(r)H_(r)P^(e)H^(e)R^(e)┘ where P^(e), R^(e) and P_(i)s, 2≦i≦r, arepermutation matrices. Furthermore, we may permute the columns of H_(M)to let R^(e) be the identity matrix. The resultant matrix would thushave a form like the expression enumerated as (13a) above. Thiscompletes the proof.

Summary of Theorems and Proofs

To summarize, the disclosed theorems and proofs of the exemplaryembodiments have the following results for a regular cycle GF(q) codewith associated graph G.

-   -   1. If G is d-regular with d=2r, r>0, Theorem 1 can be applied.    -   2. If G is d-regular with d=2r+1, r>0, and G has at least one        1-factor, Theorem 2 can be applied.

B. Nonbinary Irregular LDPC Code

Cycle codes over large Galois fields (e.g., q≧64) can achievenear-Shannon-limit performance. However, the performance gain brought byusing LDPC cycle codes over large Galois fields significantly increasesthe decoding complexity—thereby mitigating the benefits. LDPC codes oversmall to moderate Galois fields (e.g., 4≦q≦32) may be attractive from adecoding complexity point of view. Again however, a high error floor forcycle codes over GF(q) with moderate q has been observed. The high errorfloor may be caused, at least in part, by undetected errors due to thecodes' poor distance spectrum. In fact, cycle codes over small tomoderate Galois fields (e.g., between 4 and 32) suffer from performanceloss due to a “tail” in the low weight regime of the distance spectrum.See, e.g., X.-Y. Hu and E. Eleftheriou, “Binary representation of cycletanner graph GF(2b) codes,” Proc. International Conference onCommunications, vol. 27, no. 1, pp. 528-532, June 2004. In order tolower the error floor of cycle codes, exemplary embodiments of thedisclosed apparatus, systems and method employ irregular codes that aredesigned to increase the code's performance for high SNR. Theseexemplary irregular codes have an irregular column weight distributionby replacing a portion of columns of weight 2 of H by columns of weightt>2, (e.g., t=3 or t=4). This strategy can (1) increase the minimumHamming distance of the code, (2) decrease the multiplicities of lowweight codewords and/or (3) may improve the code performance at thewaterfall region due to irregular column degree distribution. In someembodiments, H has n₁ columns having weight 2 and n₂ columns havingweight t. The mean column weight may be expressed as:

$\begin{matrix}{\eta = {\frac{{2n_{1}} + {t\; n_{2}}}{n} = {2 + {\left( {t - 2} \right){\frac{n_{2}}{n}.}}}}} & \left( {14a} \right)\end{matrix}$

In order to achieve linear-time encodability (as discussed in Section 3below), n₁ can be restricted to be greater than or equal to m, that is,0≦n₂≦(n−m). Therefore, it can be said that 2≦η≦2+(t−2)r where r=(n−m)/n,and

$\begin{matrix}{{n_{1} = {n\frac{\left( {t - \eta} \right)}{t - 2}}},\mspace{14mu}{n_{2} = {\frac{n\left( {\eta - 2} \right)}{t - 2}.}}} & \left( {14b} \right)\end{matrix}$The matrix H may be arranged asH=[H ₁ |H ₂],  (15)where H₁ contains all weight 2 columns and H₂ contains all weight tcolumns. Of note, H₁ is of size m×n₁ and H₂ is of size m×n₂.

3. Properties of the Proposed Nonbinary LDPC Codes

Based on the structures presented in Section 2 above, the embodiments ofthe present disclosure which use the disclosed irregular and regularLDPC codes may have several appealing properties of normal regular cycleGF(q) codes on the encoding, decoding, and storage requirements aspects.

A. Linear-Time Encoding in Parallel

The representation in Theorems 1 and 2 enable efficient encoding asfollows. For d=2r, the codeword x can be partitioned into rsub-codewords of size m as x=[x_(c,1) ^(T),x_(c,2) ^(T), . . . , x_(c,r)^(T)]^(T). For d=2r+1, the codeword x can be portioned into r+1sub-codewords as x=[x_(c,1) ^(T),x_(c,2) ^(T), . . . , x_(c,r)^(T),x_(e) ^(T)]^(T), where x_(c,i) is one of size m, 1≦i≦r, and xe isof size m/2. Without loss of generality, it can be assumed that H ₁ isfull rank and x_(c,1) contains the parity symbols and the rest of xcontain information symbols, which leads to a code rate of (d−2)/d.Therefore, a valid codeword satisfies Hx=0, which implies that

$\begin{matrix}{{{\overset{\_}{H}}_{1}x_{c,1}} = \left\{ \begin{matrix}{{{{- P_{2}^{c}}{\overset{\_}{H}}_{2}x_{c,2}} - \ldots - {P_{r}^{c}{\overset{\_}{H}}_{r}x_{c,r}}},} & {d = {2r}} \\{{{{- P_{2}^{c}}{\overset{\_}{H}}_{2}x_{c,2}} - \ldots - {P_{r}^{c}{\overset{\_}{H}}_{r}x_{c,r}} - {P^{e}{\overset{\_}{H}}^{e}x_{e}}},} & {d = {{2r} + 1}}\end{matrix} \right.} & (16)\end{matrix}$

From the equation enumerated as (12) above, the matrix {tilde over (H)}₁is a block diagonal diag ({tilde over (H)}_(1,1) ^(c), . . . , {tildeover (H)}_(1,L) ₁ ^(c)). According to the sizes of {{tilde over(H)}_(1,l) ^(c)}_(l=1) ^(L) ¹ , x_(c,1) can be partitioned and the righthand side of the equation enumerated as (16) above into L₁ pieces as[b_(T) ¹, . . . , b_(L) ₁ ^(T)]^(T), respectively. Thus, computation ofx_(c,1) requires solving the following L₁ equations{tilde over (H)} _(1,i) ^(c) x _(c,1,i) =b _(i), 1≦i≦L ₁.  (17)

A linear time algorithm for solving these equations can be applied. See,e.g., J. Huang and J.-K. Zhu, “Linear time encoding of cycle GF(2^(p))codes through graph analysis,” IEEE Commun. Lett., vol. 10, pp. 369-371,May 2006. Specifically, to solve an equation in the form of {tilde over(H)}^(c)x=b, where x=[x₁, x₂, . . . , x_(k)]^(T), b=[b₁, b₂, . . . ,b_(k)]T, and {tilde over (H)}^(c) has the structure in the expressionenumerated as (10) above, the following algorithm may be used.z ₁ =b ₁ ; z _(i)=γ_(i-1) z _(i-1) +b _(i) , i=2, 3, . . . , k;  1.y _(k)=(1+γ₁γ₂ . . . γ_(k))⁻¹ z _(k);y _(i) =z _(i)−γ₁γ₂ . . . γ_(i-1)γ_(k) y _(k) , i=1, 2, . . . , k−1;  2.x _(i)=α_(i) ^(−l) y _(i) , i=1, 2, . . . , k.  3.

-   -   where γ_(i)=α_(i) ⁻¹β_(i), i=1, 2, . . . , k.        It can be assumed that the coefficients have been stored before        computing. The computation complexity may then be 2(k−1)        additions, 2(k−1) multiplications, and k+1 divisions over GF(q).

It is noted that solving these L₁ equations can be performed inparallel, thus encoding of exemplary embodiments can be performed inparallel in linear time. This provides flexibility in the implementationof efficient encoders, and is especially desirable when the codewordlength is large. It is also noted that the universal linear-timeencoding algorithm of for cycle codes works only in a serial manner.Fast and parallel encoding is quite desirable especially when the blocklength is large, or, when multiple rounds of encoding is needed for theproposed OFDM PAPR reduction, as will be detailed in section 5.

B. Reduction on the Storage Requirement

In prior embodiments, the storage cost for H contains two parts. Onepart corresponds to the nonzero entries of H and the other partcorresponds to the structural information for H denoted as thestructural storage cost. Compared with general cycle GF(q) codes whichdo not have the structures presented in Section 2, the structuralstorage cost for regular cycle GF(q) codes can be greatly reduced. Toperform sum-product decoding for a general cycle GF(q) code, 2n (┌logm┐+┌log n┐) bits are needed to store the row and column indices for the2n nonzero entries, where log is a base-2 logarithm operation, ┌x┐ isthe minimum integer no less than x, ┌log m┐ and ┌log n┐ bits are used tostore the row and column index for each nonzero entry of H respectively.See, e.g., M. C. Davey and D. Mackay, “Low-density parity-check codesover GF(q),” IEEE Commun. Lett., vol. 2, pp. 165-167, June 1999.Whereas, for a regular cycle GF(q) code which has a structure as in theexpression enumerated as (11) or (13a) above, not more than 2n ┌log m┐bits are needed to store the interleavers and their inversescorresponding to matrices P_(i) ^(c)s and P_(e), where ┌log m┐ bits areused to store an element for interleavers and their inverses. Thestorage cost for the parameters k_(i,1), 1≦l≦L_(i) corresponding tomatrix H _(i), 1≦i≦r, is negligible. Thus, it can be seen that comparedwith general cycle GF(q) codes the reduction of structural storage costfor regular cycle GF(q) codes is more than 50 percent. See, e.g., J.Huang, S. Zhou, J.-K. Zhu and P. Willett, “Group-theoretic analysis ofCayley-graph-based cycle GF(2p) codes,” IEEE Trans. Commun., vol. 57,no. 6, pp. 1560-65, June 2009.

C. Parallel Processing in Sequential BP Decoding

Iterative decoding based on belief propagation (BP) has receivedsignificant attention recently, mostly due to its near-Shannon-limiterror performance for the decoding of LDPC codes and turbo codes. See,e.g., R. G. Gallager, Low Density Parity Check Codes, Cambridge, Mass.:MIT Press, 1963; D. J. C. Mackay, “Good error-correcting codes based onvery sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp.399-431, March 1999; F. R. Kschischang, B. J. Frey and H. A. Loeliger,“Factor graphs and the sum-product algorithm,” IEEE Trans. Inform.Theory, vol. 47, pp. 498-519, February 2001; and C. Berrou and A.Glavieux, “Near-optimum error-correcting coding and decoding:Turbo-codes,” IEEE Trans. Commun., vol. 44, pp. 1261-1271, October 1996.Iterative decoding based on BP works on the code's Tanner-graph orfactor graph in an iterative manner through exchange of softinformation. See, e.g., R. M. Tanner, “A recursive approach to lowcomplexity codes,” IEEE Trans. Inform. Theory, vol. 27, pp. 533-547,September 1981, and F. R. Kschischang, B. J. Frey and H. A. Loeliger,“Factor graphs and the sum-product algorithm,” IEEE Trans. Inform.Theory, vol. 47, pp. 498-519, February 2001. As for LDPC codes, thereexist two kinds of processing units: variable node processing units andcheck (or constraint) node processing units corresponding to variablenodes and check nodes respectively, and two kinds of messages areexchanged between variable nodes and check nodes during iterations:variable-to-check messages and check-to-variable messages. See, e.g., J.T. Zhang and M. P. C. Fossorier, “Shuffled iterative decoding,” IEEETrans. Commun., vol. 53, pp. 209-213, February 2005. In addition, threedifferent updating schedules for BP decoding of LDPC codes can beemployed—parallel updating, sequential updating and partially parallelupdating.

Parallel Updating—In parallel updating, each iteration contains ahorizontal step followed by a vertical step. At the horizontal step, allcheck nodes update in parallel to the output check-to-variable messagesusing the input variable-to-check messages. Then, at the vertical step,all variable nodes update in parallel to the output variable-to-checkmessages using the input check-to-variable messages. The updatingschedule for standard BP is thus inherently fully parallel.

Sequential Updating—In sequential updating, a sequential version of thestandard BP is proposed to speed up the convergence of BP decoding,which is denoted as shuffled BP or sequential updating schedule. See,e.g., J. T. Zhang and M. P. C. Fossorier, “Shuffled iterative decoding,”IEEE Trans. Commun., vol. 53, pp. 209-213, February 2005, and H. Kfirand I. Kanter, “Parallel versus sequential updating for beliefpropagation decoding,” Physica A: Statistical Mechanics and itsApplications, vol. 330, pp. 259-270, December 2003. The updatingschedule for sequential BP is totally sequential—in each iteration, thehorizontal step and vertical step processes are performed jointly, butin a column-by-column manner. It has been shown through simulations thatthe average number of iterations of the sequential BP algorithm can beabout half that of the parallel BP algorithm, where parallel BP andsequential BP decoding achieve similar error performance. See, e.g., J.T. Zhang and M. P. C. Fossorier, “Shuffled iterative decoding,” IEEETrans. Commun., vol. 53, pp. 209-213, February 2005; H. Kfir and I.Kanter, “Parallel versus sequential updating for belief propagationdecoding,” Physica A: Statistical Mechanics and its Applications, vol.330, pp. 259-270, December 2003; and J. T. Zhang and M. P. C. Fossorier,“Shuffled belief propagation decoding,” in Proceedings of the 36thAsilomar Conference on Signals, Systems and Computers, vol. 1, pp. 8-15,November 2002. The complexity per iteration for both the sequential andparallel algorithms is similar, resulting in a lower total complexityfor the sequential BP algorithm. See, e.g., J. T. Zhang and M. P. C.Fossorier, “Shuffled iterative decoding,” IEEE Trans. Commun., vol. 53,pp. 209-213, February 2005; and H. Kfir and I. Kanter, “Parallel versussequential updating for belief propagation decoding,” Physica A:Statistical Mechanics and its Applications, vol. 330, pp. 259-270,December 2003.

Partially Parallel Updating—In partially parallel updating, in order todecrease the decoding delay of the sequential BP and preserve theparallelism advantages of the parallel BP, a partially parallel decodingscheme named “group shuffled BP” is developed. See, e.g., J. T. Zhangand M. P. C. Fossorier, “Shuffled iterative decoding,” IEEE Trans.Commun., vol. 53, pp. 209-213, February 2005. In the group shuffled BPalgorithm, the columns of H are divided into a number of groups. In eachgroup, the updating of messages is processed in parallel, but theprocessing of groups remains sequential. When the number of groups isone, group shuffled BP reduces to the parallel BP algorithm. But if thenumber of groups equals the number of columns of H, group shuffled BPreduces to the sequential BP algorithm. Thus, one can conclude that thegroup shuffled BP (partially parallel BP) algorithm offers betterthroughput/complexity tradeoffs in the implementation of efficientdecoders.

With respect to the sequential BP algorithm, if there are consecutivecolumns of H which are orthogonal to each other (i.e., no two columnsintersect at a common row), then the updating for these columns can becarried out simultaneously. By performing updating for consecutiveorthogonal columns simultaneously, the throughput of sequential BPalgorithm can be improved without any penalty in error performance ortotal decoding complexity. This algorithm is denoted as sequential BPdecoding with parallel processing. Sequential BP decoding with parallelprocessing is hence analogous in principle to a partially parallel BPalgorithm where the columns in each group are orthogonal.

For a cycle GF(q) code, a collection of columns of H are orthogonal ifand only if their corresponding edges in its associated graph G areindependent. With the structures presented in Section 2, orthogonalcolumns for regular cycle GF(q) codes can be easily located. Of note:

-   -   The columns of H corresponding to edges of a 1-factor of G are        orthogonal.    -   If every component of a 2-factor is an even cycle, it is defined        as an even 2-factor. Further, if a 2-factor is even, its edges        can be partitioned into two orthogonal groups. For example, the        2-factor illustrated in FIG. 3 is even, which contains one        length-2 cycle C₁=v₂e₈v₃e₁₁v₂ and one length-4 cycle        C₂=v₁e₇v₄e₃v₅e₉v₆e₄v₁. The edges of the 2-factor illustrated in        FIG. 3 can therefore be partitioned into two orthogonal groups        {e₈, e₇, e₉} and {e₁, e₃, e₄}, as illustrated in FIGS. 4 a and 4        b.    -   If a 2-factor is not even, its edges can be partitioned into        three orthogonal groups. For example, as for the 2-factor        illustrated in FIG. 7( a), which contains one length-4 cycle        C₁=v₁e₁v₂e₂v₃e₃v₄e₄v₁ and one length-5 cycle        C₂=v₅e₅v₆e₆v₇e₇v₈e₈v₉e₉v₅, its edges can be partitioned into        three orthogonal groups {e₁, e₃, e₅, e₇},{e₂, e₄, e₆, e₈} and        {e₉} as illustrated in FIG. 7( b).

Based on the aforementioned facts, the following results for d-regularcycle GF(q) codes can be summarized.

1) For a d-regular graph G with d=2r, it has r edge-disjoint 2-factors;if the number of even 2-factors is t, then edges of G can be partitionedinto 3r−t=3/2d−t orthogonal groups, 0≦t≦d/2.

2) For a d-regular graph G with d=2r+1, if it contains at least one1-factor, then it can be decomposed into r+1 edge-disjoint componentswhich consist of one 1-factor and r 2-factors; denote the number of even2-factors as t, then the edges of G can be partitioned into3r−t+1=3/2d−t−1/2 orthogonal groups, 0≦t≦(d−1)/2.

3) If the d-regular graph G is 1-factorable, then its edges can bepartitioned into d orthogonal groups.

Compared with sequential BP decoding, which works in a column-by-columnmanner and takes n steps, by running updating for columns in eachorthogonal group simultaneously, throughput of sequential BP decodingalgorithm for regular cycle GF(q) codes can be improved by a factor atleast 2n/3d. It is noted that n is usually large while d is usuallysmall. The resulting large throughput improvement may be appealing inthe implementation of efficient decoders. It is also noted that theperformance and complexity advantages of sequential BP decoding are notcompromised by this approach.

4. The Design of the Proposed Nonbinary LDPC Codes

A. Nonbinary Regular Cycle Code

In Section 2 above, the preferred structure of the parity check matrixfor regular cycle GF(q) codes was disclosed. Now, the preferred designphilosophy of regular cycle GF(q) codes is disclosed. In the preferredembodiments, a two step process to design regular cycle GF(q) codes isused. First, the code structure that specifies the locations of nonzeroentries in the check matrix is designed. The code structure is reflectedby an associated graph, which is desired to have properties known to beadvantageous—such as large girth, small diameter and good expansionproperty. See, e.g., J. Rosenthal and P. O. Vontobel, “Constructions ofLDPC codes using Ramanujan Graphs and ideals from Margulis,” inProceedings of the 38th Annual Allerton Conference on Communication,Control, and Computing, pp. 248-257, 2000; and M. Ipser and D. A.Spielman, “Expander codes,” IEEE Trans. Inform. Theory, vol. 42, pp.1710-1722, November 1996. Then, in the second step, the nonzero entriesof the parity check matrix are determined.

I. Structure Design of the Check Matrix

In exemplary embodiments of the present disclosure, at least three mainmethods to find a regular associated graph with advantageous propertiesmay be used: (1) adoption of regular graphs with good properties, suchas the Ramanujan graphs; (2) a computer search algorithm, for example,using a modified version of the progressive edge-growth (PEG) algorithm;and (3) utilize the structure results presented in Section 2 above toconstruct regular associated graphs through carefully designinginterleavers. See, e.g., X. Y. Hu, E. Eleftheriou and D.-M. Arnold,“Regular and irregular progressive edge-growth Tanner graphs,” IEEETrans. on Inform. Theory, vol. 51, January 2005; and, G. Davidoff, P.Sarnak and A. Valette, Elementary Number Theory, Group Theory, andRamanujan Graphs, Cambridge University Press, 2002.

Method 1: Code Structure Design Based on Regular Graphs.

In some embodiments, good regular graphs are used to design the codestructure, for example, the Ramanujan graphs. See, e.g., G. Davidoff, P.Sarnak and A. Valette, Elementary Number Theory, Group Theory, andRamanujan Graphs, Cambridge University Press, 2002. A d-regularRamanujan graph is defined by the property that the second largesteigen-value of its adjacency matrix is no greater than 2√d−1 and thus isknown to have good expansion properties, large girth and small diameter.In particular, the girth of Ramanujan graphs is asymptotically a factorof 4/3 better than the Erdos-Sachs bound, which in terms of girthappears to be the best d-regular graphs known.

Good known graphs may be limited in the number of code choices. Given ad-regular graph G with m vertices and girth g, if it contains at leastone 1-factor M (one 2-factor G₁, respectively), a d−1-regular(d−2-regular, respectively) graph G′ from G can be obtained by deletingthe edges of M (G₁, respectively) from G. The resultant graph G′ may bea d−1-regular (d−2-regular, respectively) graph with m vertices andgirth no less than g. Utilizing G′ as the associated graph, one canconstruct a check matrix with fixed row weight d−1 (d−2, respectively).

Method 2: Code Structure Design Based on Computer Search.

Computer search based algorithms have been adopted to construct LDPCcodes. Among them, the progressive edge-growth (PEG) algorithm has beenshown to be efficient and feasible for constructing LDPC codes withshort code lengths and high rates as well as LDPC codes with long codelengths. The PEG algorithm constructs Tanner graphs having a large girthin a best effort sense by progressively establishing edges betweensymbol and check nodes in an edge-by-edge manner. Given the number ofsymbol nodes, the number of check nodes and the symbol-node-degreesequence of the graph, an edge-selection procedure is started such thatthe placement of a new edge on the graph has as small impact on thegirth as possible. After a best effort edge has been determined, thegraph with this new edge is updated, and the procedure continues withthe placement of the next edge. Compared with other existingconstructions, the predominant advantage of PEG algorithm is that itsuccessfully generates good LDPC codes for any given block length andany rate. The PEG algorithm can also be adopted to construct regularLDPC codes which have fixed row weight and fixed column weight. See,e.g., X. Y. Hu, E. Eleftheriou and D. M. Arnold, “Regular and irregularprogressive edge-growth Tanner graphs,” IEEE Trans. on Inform. Theory,vol. 51, January 2005.

It is important to note that the PEG algorithm constructs Tanner graphswith large girth. Further, with a slight modification the PEG algorithmcan be adopted to construct associated graphs with large girth for cycleGF(q) codes, including irregular, regular and bipartite regular cycleGF(q) codes. Based on this observation, some embodiments of the presentdisclosure utilize a modified PEG algorithm to construct three kinds ofregular cycle GF(q) codes.

In exemplary embodiments, given parameters n, m, d with dm=2n, ad-regular associated graph G with m vertices and n edges can beconstructed.

1) If d=2r in the embodiment, the modified PEG algorithm can be appliedto obtain a 2r-regular graph G. With the graph G a regular cycle GF(q)code with degree 2r can be constructed.

2) If d=2r+1 (m must be even), m/2 disjoint edges in G which correspondto a 1-factor of G should be first established. Then, the modified PEGalgorithm can be applied to obtain a 2r+1-regular graph G. With thegraph G, a regular cycle GF(q) code with degree 2r+1 can be constructed.

3) If m=2m1, the modified PEG algorithm may be applied to obtain ad-regular bipartite graph G. With the graph G, a d-regular bipartitecycle GF(q) code can be constructed.

Method 3: Code Structure Design Based on the Equivalent Form of theCheck Matrix.

In another exemplary embodiment, the structure results presented insection 2 may be used as the methodology for constructing regular cycleGF(q) codes. Theorems 1 and 2 above can be used construct regular cycleGF(q) codes. For example, given the parameters n, m, d with dm=2n, aparity check matrix H with fixed row weight d and column weight 2 can beconstructed.

1) If d=2r, Theorem 1 may be applied to construct a matrix H having theform in the expression enumerated as (11) above by carefully designingthe interleavers corresponding to permutation matrices P_(i)s andappropriately choosing the quantity k_(i,l), 1≦i≦r, 1≦l≦L_(i).

2) If d=2r+1, Theorem 2 may be applied to construct a matrix H havingthe form of the expression enumerated as (13a) above by carefullydesigning the interleavers corresponding to permutation matrices P_(i)sand P^(e) and appropriately choosing the quantity k_(i,l), 1≦i≦r,1≦l≦L_(i).

II. Determination of Nonzero Entries of the Check Matrix

The selection of the nonzero entries of H affects the code performanceand therefore is an important design parameter. As a point of analysis,it is assumed that a binary extension filed, that is, q=2^(p) for somep. However, the following results can be generalized straightforward toother Galois fields. It may be assumed that ξ is a primitive element ofGF(2^(p)) satisfying f(ξ)=0, where f(x)=x^(p)+f^(p)−1x^(p)−1+ . . . +f₀is a primitive polynomial of degree p over GF(2). Further, it may beassumed that Z_(q-1) be the additive group modulo q−1. The mappingM:GF(q)\{0}→

⁻¹, ξ^(i) →i, i=0, 1, . . . , q−2,  (18)is therefore an isomorphism from the multiplicative group of GF(q) toZ_(q-1).

The sub-matrix associated with a length-k cycle is equivalent to {tildeover (H)}^(c) as shown in the expression enumerated as (10) above. It isknown that the cycle is irresolvable if and only if {tilde over (H)}^(c)is full-rank, i.e., Π_(i=1) ^(k)α_(i) ⁻¹β_(i)≠1. See, e.g., J. Huang andJ.-K. Zhu, “Linear time encoding of cycle GF(2p) codes through graphanalysis,” IEEE Commun. Lett., vol. 10, pp. 369-371, May 2006. If thegain of the edge e_(i) is defined as γi=α_(i) ⁻¹β_(i), then {tilde over(H)}^(c) is full-rank if and only if Π_(i=1) ^(k)γ_(i)≠1, i.e.,

$\begin{matrix}{{\sum\limits_{i = 1}^{k}{\mathcal{M}\left( \gamma_{i} \right)}} \neq {0\mspace{14mu}{\left( {{{mod}\mspace{14mu} q} - 1} \right).}}} & (19)\end{matrix}$

It has been observed that resolvable cycles with short length correspondto low-weight codewords, which may induce undetected errors during thedecoding process. See, e.g., J. Huang, S. Zhou, J.-K. Zhu and P.Willett, “Group-theoretic analysis of Cayley-graph-based cycle GF(2p)codes,” IEEE Trans. Commun., vol. 57, no. 6, pp. 1560-65, June 2009. Toachieve good decoding performance, exemplary embodiments include codesdesigned with the following design criterion:

C1: choose nonzero entries of the check matrix to make as many cyclesirresolvable as possible, especially those having short length.

Based on the associated graph, all the cycles can be found. Then, anappropriate γ_(i) may be chosen through solving a set of inequalities(e.g., the expression enumerated as (19) above) corresponding to thosecycles of short length. Given γ_(i) for an edge e_(i), the value α_(i)can be randomly generated with uniform distribution, and the value β_(i)can be determined using βi=α_(i)γ_(i). This exemplary algorithm appliesto both regular cycle GF(q) codes and irregular cycle GF(q) codes.

B. Nonbinary Irregular LDPC Code

In exemplary embodiments, H₁ and H₂ may be designed to maximally benefitfrom the structure developed for regular cycle code in Section 2 above.In one embodiment, this is accomplished by noting that H₁ corresponds tothe check matrix of a general cycle code and designing H₁ to be as closeto a regular cycle code as possible. Specifically, the matrix may besplit asH=[H _(1a) |H _(1b) |H ₂],  (20)where the matrix H_(1a), is of size m×n_(1a) and the matrix H_(1b) is ofsize m×n_(1b). The number n_(1a) can be the largest integer not greaterthan n₁ that can render d_(1a)=(2n_(1a))/m an integer—that is, H_(1a),is the largest sub-matrix of H₁ that could be made d_(1a)-regular.Further, if n_(1a)=n₁, then n_(1b)=0. As such, H₁ itself can be maderegular, which is a special case.

The detailed design steps of an exemplary design method include:

-   -   Step 1: Specify the structure of H_(1a). Construct a cycle code        of fixed row weight d_(1a) using the design methodologies        outlines above with respect to regular cycle codes. See,        e.g., J. Huang, S. Zhou, and P. Willett, “Structure, Property,        and Design of Nonbinary Regular Cycle Codes,” IEEE Trans. on        Communications, vol. 58, no. 4, April 2010.    -   Step 2: Specify the structure of H_(1b) and H₂. Apply the        progressive edge-growth (PEG) algorithm to attach n_(1b) columns        of weight 2 and n₂ columns of weight t to the matrix H_(1a).        See, e.g., X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, “Regular        and irregular progressive edge-growth tanner graphs,” IEEE        Trans. Inform. Theory, vol. 51, no. 1, pp. 386-398,        January 2005. In this way, the structure of H in the expression        enumerated as (20) above is established.    -   Step 3: Specify the non-zero entries of H₁. The submatrix        H₁=[H_(1a)|H_(1b)] can be regarded as a check matrix of a cycle        code. Hence, design criterion can be applied to choose        appropriate nonzero entries for H₁ to make as many as possible        short length cycles of the associated graph of H₁ irresolvable.        See, e.g., J. Huang and J.-K. Zhu, “Linear time encoding of        cycle GF(2p) codes through graph analysis,” IEEE Commun. Lett.,        vol. 10, pp. 369-371, May 2006.    -   Step 4: Specify the non-zero entries of H₂. The nonzero entries        of H₂ are generated randomly with a uniform distribution over        the set GF(q)\0.

The proposed nonbinary irregular LDPC codes attempt to make a largeportion of its check matrix into a regular cycle code. In this way, manybenefits from regular cycle codes can be retained. FIG. 5 compares theperformance of irregular LDPC codes over GF(16) with different meancolumn weights. All the codes have rate of ½ and block length of 1008bits. More specifically, FIG. 5 shows a performance comparison ofirregular codes over GF(16) with different mean column weights t=3, r=½,and the block length is 1008 bits, and for the η=2.0 and η=2.2 cases,the probability of undetected errors, which contributes to the errorfloor of the block error rate, is also plotted. BPSK modulation is usedon the binary input AWGN channel and the decoder uses the sequential BPalgorithm with a maximum of 80 iterations. See, e.g., H. Kfir and I.Kanter, “Parallel versus sequential updating for belief propagationdecoding,” Physica A: Statistical Mechanics and its Applications, vol.330, pp. 259-270, December 2003; and J.-T. Zhang and M. P. C. Fossorier,“Shuffled iterative decoding,” IEEE Trans. Commun., vol. 53, pp.209-213, February 2005. As can be seen from FIG. 5, it is noted that thecodes with η=2.0 and η=2.2 show an error floor above 10⁻⁵ which arecaused by undetected errors. No error floor above 10⁻⁵ shows if η≧2.4.Further, no undetected errors have been observed for η≧2.4 in oursimulations. In reference to FIG. 5, it is further noted that as ηincreases from 2.4 to 2.6 and 2.8, the code performance degrades.Therefore, the code with η=2.4 may be considered the optimum one in thissetting. FIG. 5 also shows the performance comparison between theirregular LDPC codes over GF(16) with binary optimized LDPC code. Theperformance of Mackay's (3,6)-regular code and cycle codes over GF(64)and GF(256) are also included. See, e.g., J. Huang, S. Zhou, and P.Willett, “Structure, Property, and Design of Nonbinary Regular CycleCodes,” IEEE Trans. on Communications, vol. 58, no. 4, April 2010. Itcan be further seen from FIG. 5 that by adopting an irregular columnweight distribution, the code's performance has been greatly improved.

5. Peak-to-Average Power Ratio Reduction of the Proposed Nonbinary LDPCCodes

One major problem associated with OFDM is the high peak-to-average powerratio (PAPR), which can be defined as

$\begin{matrix}{{{PAPR}:=\frac{\max\left( {{x(t)}}^{2} \right)}{E\left\lbrack {{x(t)}}^{2} \right\rbrack}},} & (21)\end{matrix}$where x(t) is the transmitted OFDM signal. PAPR can be evaluated ateither baseband or passband, depending on the choice of x(t). See, e.g.,S. Litsyn, Peak Power Control in Multicarrier Communications, CambridgeUniversity Press, 2007. Nonlinear amplification may cause intermodulation among subcarriers and undesired out-of-band radiation. Intheory, to limit nonlinear distortion, the amplifier at the transmittershould operate with large power back-offs.

Various PAPR reduction methods have been proposed for radio OFDMsystems. See, e.g., S. Litsyn, Peak Power Control in MulticarrierCommunications, Cambridge University Press, 2007. The preferredembodiments of the present disclosure utilize the selected mapping (SLM)approach. See, e.g., R. Bauml, R. Fischer, and J. Huber, “Reducing thepeak-to-average power ratio of multicarrier modulation by selectedmapping,” Electron. Lett., vol. 32, no. 22, pp. 2056-2057, October 1996;and M. Breiling, S. Muller-Weinfurtner, and J.-B. Huber, “SLM peak-powerreduction without explicit side information,” IEEE Commun. Lett., vol.5, no. 6, pp. 239-241, June 2001. In SLM, the transmitter generates aset of sufficiently different candidate signals which all represent thesame information and selects the one with the lowest PAPR fortransmission. In the original SLM approach, side information on whichsignal candidate has been chosen needs to be transmitted and can causesignaling overhead. In addition, side information has high importanceand should be strongly protected. In the currently preferred approach,some additional bits, used to select different scrambling code patterns,are inserted into the information bits, before applying scrambling andchannel encoding. In this way, the side information bits are containedin the data and do not require separate encoding.

The fact that the generator matrix G of a LDPC code has high density iswell known, but rarely utilized. In some embodiments, this property ofLDPC is used to reduce PAPR, following the principle of SLM. See, e.g.,M. Breiling, S. Muller-Weinfurtner, and J.-B. Huber, “SLM peak-powerreduction without explicit side information,” IEEE Commun. Lett., vol.5, no. 6, pp. 239-241, June 2001. The transmitter can be said to operateas follows:

-   -   For each set of information bits to be transmitted within one        OFDM symbol, reserve z bits for PAPR reduction purpose.    -   For each choice of the values of these z bits, carry out LDPC        encoding and OFDM modulation, and calculate the PAPR.    -   Out of 2^(z) candidates, select the OFDM symbol with the lowest        PAPR for transmission.

Compared with the original SLM approach, the proposed method bypassesthe scrambling operation at the transmitter and the descramblingoperation at the receiver. Due to the non-sparseness of G, single bitchange will lead to a drastically different codeword after LDPCencoding. See, e.g., D. Mackay, Information Theory, Inference, andLearning Algorithms, Cambridge University Press, 2003. Since z is verysmall, the reduction on transmission rate is negligible. At the receiverside, those z bits are simply dropped after channel decoding. The maincomplexity increase is hence on the transmitter. Fast encoding aspresented in Section 3 is thus very important for the proposed approach.

As an example, for a systematic nonbinary LDPC code with size n×k, therecan be said to be a k×k identity matrix contained in G. Therefore, everyinformation bit change can only cause significant changes on the (n−k)parity symbols. Also, for low rate transmissions, systematic LDPC mayachieve decent PAPR reduction. However, for high rate transmissionswhere (n−k) is small, nonsystematic LDPC codes may be preferred oversystematic codes for PAPR reduction.

One exemplary way to construct a nonsystematic code from a systematiccode is as follows. The z reserved bits may be placed into the last sinformation symbols of the block u, where s=[z/p].

A matrix V can be constructed as

$\begin{matrix}{V = \begin{bmatrix}I_{{k - s}} & B \\0 & A\end{bmatrix}} & (22)\end{matrix}$where A is an invertible square matrix of size s×s and B is of size(k−s)×s. Then, the generator matrix of the nonsystematic code may beconstructed from that of a systematic code asG _(non) =G _(sys) V.  (23)

The output codeword can then be expressed as x=G_(non)u=G_(sys)Vu, whichmeans that the information block u is scrambled by the matrix V beforebeing passed to the systematic encoder. At the decoder, an estimate ofVu may be recovered, and then û obtained by left multiplying the inverseof V as

$\begin{matrix}{V^{- 1} = {\begin{bmatrix}I_{k - s} & {{- B}\; A^{- 1}} \\0 & A^{- 1}\end{bmatrix}.}} & (24)\end{matrix}$

It is noted that the size of A is very small. For example, if z=4, thens=2 when using an LDPC code over GF(4), and s=1 when using an LDPC codeover GF(16). Therefore, left multiplication of V⁻¹ has low complexityand can be done in parallel.

With certain OFDM parameters, and where each OFDM block has 1024subcarriers out of which 672 subcarriers are used for data transmission,it is possible to simulate the baseband OFDM signals with a samplingrate 4 times of the bandwidth to evaluate the complementary cumulativedistribution function (ccdf), Pr(PAPR>x). The PAPR ccdf curves for mode2 of Table I are shown in FIG. 17 for z=0, z=2, and z=4, respectively,where the corresponding curves using a 64-state rate-½ convolutionalcode (with generators) are also included. It is noted that the generatormatrix of convolutional code has low density, as each bit can onlyaffect subsequent bits within the constraint length. For convolutionalcodes, the z reserved bits are distributed uniformly among theinformation bit sequence. It can be observed from FIG. 17 that using anonbinary LDPC code with 4 bits overhead can achieve about 3 dB gainthan the case with no overhead at the ccdf value of 10⁻³. Compared withconvolutional codes using 4 bits overhead, nonbinary LDPC code with 4bits overhead can achieve about 2 dB gain at the ccdf value of 10³.Further, scrambling can be used together with convolutional codes toimprove the PAPR characteristic. See, e.g., M. Breiling, S.Muller-Weinfurtner, and J.-B. Huber, “SLM peak-power reduction withoutexplicit side information,” IEEE Commun. Lett., vol. 5, no. 6, pp.239-241, June 2001. However, scrambling is not necessary with LDPCcodes. With rate 1/2, it can be seen that systematic and nonsystematiccodes have similar PAPR reduction performance. In fact, FIG. 18 showsthat nonsystematic LDPC codes have better PAPR reduction than systematiccodes when the code rate is increased to ¾.

6. Simulation Results of the Proposed Nonbinary LDPC Codes

In this section, simulations of some embodiments were conducted toevaluate the performance of the irregular and regular LDPC GF(q) codes.

In the following simulations the codewords were transmitted over AWGNchannel with binary phase-shift-keying (BPSK) modulation. Each SNRsimulations were run until more than 40 block errors were observed or upto 1,000,000 block decodings.

Test Case 1 (Regular Versus Irregular Cycle GF(q) Codes)

FIG. 8 shows a comparison of the performance of regular and irregularcycle GF(q) codes under standard BP decoding up to 80 iterations wherethe code rate is ½ and the codeword length is 1008 bits. The cycle codesover GF(2⁶) have a symbol length of 84 and the cycle codes over GF(2⁸)have a symbol length of 63. For GF(2⁶) a bipartite regular cycle codewas also constructed. The check matrices of irregular cycle GF(q) codeswere constructed by the PEG algorithm. The check matrices of regular andbipartite regular cycle GF(q) codes were also constructed by themodified PEG algorithm described in Section 4. Nonzero entries of thecheck matrices for all cycle GF(q) codes are randomly generated with auniform distribution. Also plotted is the performance of a binaryirregular rate-½ LDPC code constructed by the PEG algorithm and that ofa rate-½ MacKay's regular-(3,6) code, both having a code length of 1008bits and decoded by standard BP up to 80 iterations. The binaryirregular code has a density-evolution-optimized degree distributionpair achieving an impressive iterative decoding threshold of 0.3347 dB,i.e. the symbol-node edge distribution is0.23802x+0.20997x²+0.03492x³+0.12015x⁴+0.01587x⁶+0.00480x¹³+0.37627x¹⁴and the check-node edge distribution is 0.98013x⁷+0.01987x⁸. See, TableII in T. Richardson, A. Shokrollahi and R. Urbanke, “Design of provablygood low-density parity-check codes,” IEEE Trans. Inform. Theory, vol.47, pp. 619-637, February 2001.

It has been shown that irregular cycle codes over GF(q) can outperformbinary degree-distribution-optimized LDPC codes. See, e.g., X.-Y. Hu andE. Eleftheriou, “Binary representation of cycle Tanner-graph GF(2b)codes,” IEEE International Conference on Communications, vol. 27, no. 1,pp. 528-532, June 2004. As shown in FIG. 8, it is noted that the regularcycle codes can also outperform binary degree-distribution optimizedLDPC codes. In fact, FIG. 8 shows that regular cycle codes and irregularcycle codes have similar performance. Of note, the error floor appearsearlier for the bipartite-graph based cycle code over GF(2⁶) than theregular and irregular cycle codes over GF(2⁶), which may be due to alarge portion of undetected errors of weight 6 corresponding to length-6resolvable cycles in its associated graph. In some embodiments, thiserror floor can be effectively lowered by careful selection of non-zeroentries in the check matrix, as will be elaborated in Test Case 3.

Test Case 2 (Sequential Versus Parallel BP Decoding)

FIGS. 9 and 10 show the comparisons on the error performance and theaverage number of iterations between the proposed sequential BP decodingwith parallel processing and standard BP decoding for those regularcycle GF(q) codes shown in FIG. 8. The maximum number of iterations wasset to be 80. Of note, as shown by FIG. 9 the sequential BP decodingwith parallel processing achieves slightly better performance than thestandard parallel BP decoding. More importantly, FIG. 10 shows that theaverage number of iterations for the sequential BP decoding is about 30percent less than that of the standard BP decoding at high SNR. Hence,the total decoding complexity for the proposed algorithm is 30 percentless than that for standard BP decoding algorithm. Moreover, theproposed parallel processing enables a speedup on the throughput ofsequential BP decoding by a factor at least 2n/3d=10.5 for the regularGF(2⁸) code and at least 2n/3d=14 for the regular and bipartite regularGF(2⁶) codes.

Test Case 3 (Determination of Nonzero Entries of the Check Matrix)

FIG. 11 shows the performance improvement for an exemplary embodimentwhen the design criterion C1 is applied to select the nonzero entries ofthe check matrix for the bipartite-graph based cycle code over GF(2⁶) inFIG. 8. The girth of the code's associated graph is 4 and it has beenfound that all of its cycles are of length 4, 6, 8 and 10. Solutions tosatisfy all inequalities (e.g., the expression enumerated as (19) above)for cycles of length 4, 6, 8, and even 10 may be searched for using arandom search. For the ‘Opt-1’ code in FIG. 11, all cycles of length 4and 6 were rendered irresolvable. For the ‘Opt-2’ code in FIG. 11, allcycles of length 4, 6, and 8 were rendered irresolvable. Thus, FIG. 11confirms that the proposed design criterion C1 can effectively lower theerror floor for cycle GF(q) codes.

Test Case 4 (Codes Constructed Through Interleaver Design Vs. CodesConstructed by PEG)

FIG. 12 shows a comparison of performance of regular cycle GF(2⁶) codesconstructed from interleaver design with a cycle GF(2⁶) code constructedby the PEG algorithm. Semi-random interleavers were used in theembodiment. The proposed sequential BP with parallel processing was usedfor decoding regular cycle codes where the sequential BP for decodingthe PEG constructed code was adopted. The maximum number of iterationswas set to be 80. The code rate was ½ and the information symbol lengthwas 112 symbols over GF(2⁶). The associated graph of ‘Code2’ iscomprised of two edge-disjoint spanning cycles of length 112 and theassociated graph of ‘Code1’ is comprised of two edge-disjoint 2-factors,where each 2-factor consists of 16 disjoint cycles of length 7. For thecodes labeled with ‘Optimized’ the design criterion C1 to chooseappropriate nonzero entries for the check matrices was applied. It canbe seen from FIG. 12 that, compared with codes constructed by the PEGalgorithm, the performance loss of regular cycle codes constructed usingsemi-random interleavers is only 0.15 dB at block-error-rate of 10⁻⁴. Itis noted that careful interleaver design could further improveperformance.

Other embodiments were simulated for performance analysis purposes usingboth an AWGN channel (Ĥ[k]=1,∀k in the expression enumerated as (4)above) and an underwater Rayleigh fading channel. Specifically, thebandwidth was 12 kHz, and the channel delay spread is 10 ms, resultingin 120 channel taps in discrete-time. Equal-variance complex Gaussianrandom variables were used on each tap.

The two channel models are significantly different—one without channelfading and the other with multipath fading from a rich scatteringenvironment. The coding performance based on these two different channelmodels was compared to facilitate code selection. It is also noted thatpractical underwater acoustic channels could be far more complex, e.g.,with sparse multipath structure and much longer impulse response.

When the LDPC coding alphabet is matched to the modulation alphabet,i.e., p=b, or when p is an integer multiple of b, constellation labelingdoes not affect the error performance of the proposed system. Further,interleaving the codeword means a column rearrangement of the code'sparity check matrix, implying that interleaving can be absorbed into thecode design and does not need to be considered explicitly. In thefollowing simulation results, Gray labeling and identity interleaversare used.

OFDM parameters were used as well. See, e.g., B. Li, S. Zhou, M.Stojanovic, L. Freitag, and P. Willett, “Multicarrier communication overunderwater acoustic channels with nonuniform Doppler shifts,” IEEE J.Oceanic Eng., vol. 33, no. 2, April 2008 and B. Li, S. Zhou, M.Stojanovic, L. Freitag, J. Huang, and P. Willett, “MIMO-OFDM over anunderwater acoustic channel,” in Proc. MTS/IEEE OCEANS conference,Vancouver, BC, Canada, Sep. 29-Oct. 4, 2007. Each OFDM block is ofduration 85.33 ms, and has 1024 subcarriers, out of which 672subcarriers are used for data transmission and each OFDM block containsone codeword. The FFTQSPA algorithm is used for nonbinary LDPC decoding,where the maximum number of iterations is set to 80.

Test Case 5 (Combination of Coding and Modulation)

FIGS. 19 and 20 show a comparison of the error performance of differentexemplary coding and modulation combinations under the AWGN and Rayleighfading channels, respectively. The following observation can be made:

-   -   A QPSK system with rate ⅞ coding over GF(16) leads to a data        rate of 1.75 bits/symbol while a 16-QAM system with rate ½        coding over GF(16) and an 8-QAM system with rate ⅔ coding over        GF(8) leads to a data rate of 2 bits/symbol. As seen in FIG. 19,        the three systems achieved similar performance over the AWGN        channel. However, as seen from FIG. 20, the QPSK system with        rate ⅞ coding (and the 8-QAM system with rate ⅔ coding) is about        4 dB (1.3 dB) worse than the 16-QAM system with rate ½ over the        Rayleigh fading channel at BLER of 10⁻².    -   A 64-QAM system with rate ⅔ coding has a data rate of 4        bits/symbol, while a 16-QAM with rate ⅚ (⅞) coding has data rate        of 3.34 (3.5) bits/symbol. As seen from FIG. 19, the 16-QAM        system with rate ⅚ coding (and the 16-QAM system with rate ⅞        coding) achieves about 5.7 dB (5 dB) gain against the 64-QAM        system with rate ⅔ coding at BLER of 10⁻² over the AWGN channel.        However, as seen from FIG. 20, the 16-QAM system with rate ⅚        coding has similar performance as the 64-QAM system with rate ⅔        coding over the Rayleigh fading channel, and the 16-QAM system        with rate ⅞ coding is about 2 dB worse than the 64-QAM system        with rate ⅔ coding over the Rayleigh fading channel at BLER of        10⁻².

Hence, it is proposed that different coding and modulation combinationswith a similar data rate could have quite different behaviors in theAWGN and Rayleigh fading channels. Without being bound by any theory, itis theorized that this effect may be due to the fact that differentperformance metrics matter for AWGN and Rayleigh fading channels. See,D. Divsalar and M. K. Simon, “The design of trellis coded MPSK forfading channels: performance criteria,” IEEE Trans. Commun., vol. 36,no. 9, pp. 1004-1012, September 1988. Specifically, minimum Hammingdistance may play a significant role for the Rayleigh fadingchannel—while minimum Euclidean distance may play a significant role forthe AWGN channel. In general, a combination of low rate code and largeconstellation can yield a larger Hamming distance than that of high ratecode and small constellation, when the same spectral efficiency isachieved.

The performance of many different combinations of modulations such asBPSK, QPSK, 8-QAM, 16-QAM and 64-QAM, and LDPC codes of rate ½, ⅔, ¾, ⅚and ⅞ were simulated. For LDPC codes over GF(q) where q<64, differentcombinations of value t (3 or 4) and η (range from 2.0 to 3.0) have beensimulated. For LDPC codes over GF(64), exemplary nonbinary regular cyclecodes from are used. For the bandwidth efficiency ranging from 0.5 to 5bits/symbol, we only kept the combination that results in goodperformance in the Rayleigh fading channel and record the LDPC codeparameters. It can be seen from Table I that low-rate codes (i.e., rate½) are preferable.

TABLE I NONBINARY LDPC CODES DESIGNED FOR UNDERWATER SYSTEM. η STANDSFOR MEAN COLUMN WEIGHT. EACH CODEWORD HAS 672b BITS WITH A SIZE-2^(b)CONSTELLATION. Bits Per Code Galois Mode Symbol Rate η t FieldConstellation 1 0.5 ½ 2.8 4 GF(4) BPSK 2 1 ½ 2.8 4 GF(4) QPSK 3 1.5 ½2.8 4 GF(8)  8-QAM 4 2 ½ 2.3 3 GF(16) 16-QAM 5 3 ½ 2.0 — GF(64) 64-QAM 64 ⅔ 2.0 — GF(64) 64-QAM 7 5 ⅚ 2.0 — GF(64) 64-QAM

Test Case 6 (Performance of Different Modes)

FIGS. 13 a and 13 b show the block error rate (BLER) and bit error rate(BER) performance of all the modes in Table I over an AWGN channel. Alsoincluded are the uncoded BER curves for different modulations. FIGS. 14and 15 show the BLER and BER performance of all the modes in Table Iover OFDM Rayleigh fading channel respectively.

Also included in FIGS. 13 a, 13 b, 14 and 15 are uncoded BER curves fordifferent modulations or constellations. It can be seen that as long asuncoded BER is somewhat below 0.1, the coding performance improvesdrastically, approaching the waterfall behavior.

Test Case 7 (Comparison with CC Based BICM)

FIGS. 16 and 21 show a comparison between the performance of abit-interleaved coded-modulation (BICM) system based on a 64-staterate-½ convolutional code and the proposed nonbinary LDPC coding systemunder different modulation schemes over the OFDM Rayleigh fadingchannels, respectively. Gray labeling, random bit-level interleaver, andsoft decision Viterbi decoding are used in the test BICM system. It canbe seen from FIGS. 16 and 21 that compared with the BICM system usingthe convolutional code, nonbinary LDPC codes achieve several decibels(varying from 2 to 5 dB) performance gain at BLER of 10⁻². It is notedthat the performance of BICM may be considerably improved by using morepowerful binary codes such as turbo codes and binary LDPC codes, andthrough iterative constellation demapping. See, e.g., X. Li and J. A.Ritcey, “Bit-interleaved coded modulation with iterative decoding,” IEEECommun. Lett., vol. 1, no. 6, pp. 169-171, November 1997.

8. Test Results with Real Data

Proposed nonbinary regular and irregular LDPC codes for severalunderwater experiments have been used and the test results have beenrecorded and analyzed. See, e.g., B. Li, S. Zhou, M. Stojanovic, L.Freitag, J. Huang, and P. Willett, “MIMO-OFDM over an underwateracoustic channel,” in Proc. Of MTS/IEEE OCEANS conference, Vancouver,Canada, Sep. 30-Oct. 4, 2007; and B. Li, S. Zhou, J. Huang, and P.Willett, “Scalable OFDM design for underwater acoustic communications,”in Proc. of Intl. Conf. on ASSP, Las Vegas, Nev., Mar. 3-Apr. 4, 2008.In all experimental settings with nonbinary regular and irregular LDPCcodes of the exemplary embodiments, nearly error-free performance wasachieved. In fact, whenever the uncoded BER is below 0.1, decodingerrors for rate 1/2 codes in the experiments were not observed. Thisfinding is consistent with FIGS. 13 a-15. Hence, the goal of OFDMdemodulation can be summarized as achieving an uncoded BER to be withinthe range of 0.1 and 0.01, and therefore the coding will boost thesystem performance.

A. Field Test Results from Experiments at AUV Fest 2007 and BuzzardsBay, 2007

Nonbinary LDPC codes have been applied in a multicarrier system and datahas been collected from experiments at AUV Fest, Panama City, Fla., June2007, and at Buzzards Bay, Mass., August 2007. The detailed descriptionof the experiments can be seen in B. Li, S. Zhou, J. Huang, and P.Willett, “Scalable OFDM design for underwater acoustic communications,”in Proc. of Intl. Conf. on ASSP, Las Vegas, Nev., Mar. 30-Apr. 4, 2008,the entire contents of which is hereby expressly incorporated byreference herein.

In the AUV Fest, the sampling rate was 96 kHz. Signals with threedifferent bandwidths, (3 kHz, 6 kHz, and 12 kHz, and centered around thecarrier frequency 32 kHz) were used. The transmitter was about 9 m belowa surface buoy. The receiving boat had an array in about 20 m depthwater and the array depth was about 9 m to the top of the cage. Below,the results are reported with a transmission distance of about 500 m andthe channel delay spread of about 18 ms.

In the Buzzards Bay test, the sampling rate was 400 kHz. Signals withtwo different bandwidths, 25 kHz and 50 kHz, centered around the carrierfrequency 110 kHz, were used. The transmitter gear was deployed to thedepth of about 6 m to about 7.6 m with a water depth about 14.3 m. Thereceiver array was deployed to the depth of about 6 m with a water depthabout 14.3 m and an array spacing of about 0.2 m. Below, the results arereported with a transmission distance of about 180 m and a channel delayspread of about 2.5 ms. In both experiments, mode 2 (QPSK) and mode 4(16-QAM) listed in Table I were adopted for nonbinary LDPC coding. Inaddition, included are signal sets with convolutional coding, where a16-state rate ½ convolutional code with the generator polynomial (23,35)was used.

With QPSK modulation and rate ½ coding, the achieved spectral efficiencyafter accounting for various overheads was about 0.5 bits/sec/Hz,leading to data rates from 1.5 kbps to 25 kbps with different bandwidthsfrom 3 kHz to 50 kHz. With 16-QAM modulation and rate ½ coding, theachieved spectral efficiency was about 1 bits/sec/Hz, leading to datarates from 12 kbps to 50 kbps with different bandwidths from 12 kHz to50 kHz.

1) BER Performance for QPSK

BER results for convolutional codes (CC) with QPSK were collected andare shown in Table II, and those for the LDPC codes were collected andare shown in Table III. A total of 43008 information bits weretransmitted in each setting. In some cases, there was no decodingerror—even with a single receiver. Further, for all the cases tested,when signals from two receivers were properly combined there were noerrors after channel decoding.

2) BER Performance for 16-QAM

FIG. 22 shows the resultant BER values after channel decoding when16-QAM was used. A total of 43008 information bits were transmitted ineach setting. For the B=12 kHz case from the AUV Fest experiment, tworeceivers were needed for zero BER for LDPC, while four receivers wereneeded for zero BER for CC. For the B=25 kHz case from the Buzzards Baytest, two receivers were needed for zero BER for LDPC, while threereceivers were needed for zero BER for CC. For the B=50 kHz case fromthe Buzzards Bay test, three receivers were needed for zero BER forLDPC, while for CC, a large BER still occurred with four receivers.Without being bound by any theory, it is believed that this phenomenonmay have occurred because the nonbinary LDPC code has much bettererror-correction capability than the convolutional code used.

TABLE II BER RESULTS FOR CC WITH QPSK 1 receiver 2 receivers Bandwidth Buncoded/coded uncoded/coded AUV Fest, 3 kHz 0.1219/0.0403 0.0395/0 AUVFest, 6 kHz 0.0762/0.0063 0.0218/0 AUV Fest, 12 kHz 0.0752/0.00480.0185/0 Bay Test, 25 kHz 0.0016/0 — Bay Test, 50 kHz 0.0834/0.01910.0243/0

TABLE III BER RESULTS FOR LDPC WITH QPSK 1 receiver 2 receiversBandwidth B uncoded/coded uncoded/coded AUV Fest, 12 kHz 0.0613/0 — Baytest, 25 kHz 0.0015/0 — Bay test, 50 kHz 0.1828/0.1851 0.1102/0

B. Field Test Results from the RACE08 Experiment

A Rescheduled Acoustic Communications Experiment (RACE) took place inNarragansett Bay, R.I., from Mar. 1 through Mar. 17, 2008. The waterdepths were in the range from about 9 to about 14 meters. The primarysource for acoustic transmissions was located approximately 4 metersabove the bottom. Three receiving arrays, one at about 400 meters to theeast from the source, one at about 400 meters to the north from thesource, and one at about 1000 meters to the north from the source, werelocated with the bottom of the arrays about 2 meters above the seafloor. The arrays at about 400 meters range were 24 element verticalarrays with a spacing of 5 cm between elements. The array at the about1000 meter range was a 12 element vertical array with 12 cm spacingbetween elements. The sampling rate was fs=39.0625 kHz. The signalbandwidth was set as B=fs/8=4.8828 kHz, centered around the carrierfrequency fc=11.5 kHz. Also, K=1024 subcarriers were used, which lead toa subcarrier spacing of Δf=4.8 Hz and the OFDM duration of T=209.7152ms. The guard interval between consecutive OFDM blocks was Tg=25 ms. Thetransmission modes 2 to 5 were tested and are listed in Table I.

Our transmission file contained four packets. The first packet contained36 OFDM blocks with QPSK modulation (Mode 2), the second packetcontained 24 OFDM blocks with 8-QAM modulation (Mode 3), the thirdpacket contained 18 OFDM blocks with 16-QAM modulation (Mode 4) and thefourth and last packet contained 12 OFDM blocks with 64-QAM modulation(Mode 5). Each packet has 24192 information bits regardless of thetransmission mode. Accounting for the overheads of guard intervalinsertion, channel coding, pilot and null subcarriers, the spectralefficiency can be expressed as:

$\begin{matrix}{\beta = {{\frac{T}{T + T_{g}} \cdot \frac{672}{1024} \cdot \frac{1}{2} \cdot \log_{2}}M\mspace{14mu}{bits}\text{/}\sec\text{/}{{Hz}.}}} & (25)\end{matrix}$

From this expression, the spectral efficiencies for the RACE08experiment are 0.5864, 0.8795, 1.1727, and 1.7591 bits/sec/Hz, fortransmission modes with QPSK, 8-QAM, 16-QAM, and 64-QAM constellations,respectively. Thus, the achieved data rates are 2.86, 4.29, 5.72, and8.59 kbps, respectively. During the experiment, each transmission filewas transmitted twice every four hours, leading to 12 transmissions perday. A total of 124 data sets were successfully recorded on each arraywithin 13 days from the Julian date 073 to the Julian date 085.

The performance results on the array at 400 m to the east and on thearray at 1000 m to the north are provided herein. The channel delayspreads were around 5 ms for both settings. FIGS. 23 and 24 depict theBER and BLER after channel decoding as a function of the number ofreceiver-elements, averaged over all the data sets collected from 13days. Hence, each point in FIGS. 23 and 24 corresponds to transmissionsof 124×24192≈3.0·10⁶ information bits. FIGS. 25 and 26 plot the uncodedand coded BERs for each recorded data set at the array at 1000 m to thenorth across the Julian dates, for 16-QAM and 64-QAM constellations,respectively. It is noted that with 8 receiver-elements, error freeperformance was achieved during the 13 day operation for QPSKtransmissions. Also, very good performance was achieved for 8-QAM and16-QAM transmissions, as the BLER is below 10⁻²—which may satisfy therequirement of a practical system. Further, the average BLER wasactually below 0.1 for 64 QAM constellation. A closer look at FIG. 26shows that error-free transmissions were achieved for a large majorityof transmissions. As a result, this experiment demonstrates that theproposed transmission modes are fairly robust to the varying channelconditions within those 13 days.

In summary, nonbinary LDPC coding has been applied in multicarrierunderwater systems, where the focus was on matching the code alphabetwith the modulation alphabet. The real data shows that whenever theuncoded BER is below 0.1, normally no decoding errors will occur for therate ½ of the nonbinary LDPC codes used. This result is consistent withthe simulation results in FIGS. 13 a, 13 b, 14 and 15, as the curves atthe waterfall region are steep. The uncoded BER can serve as a quickperformance indicator to assess how likely the decoding will succeedand, therefore, the goal of an OFDM receiver design may be to achieve anuncoded BER within the range of 0.1 and 0.01—as nonbinary LDPC codingwill boost the overall system performance afterwards.

CONCLUSIONS

In preferred embodiments, apparatus, systems and methods of UWAcommunication are provided that include nonbinary regular low-densityparity-check (LDPC) cycle codes if the constellation is large (e.g.,modulation of at least 64-QAM or a Galois Field of at least 64) andnonbinary irregular LDPC codes if the constellation is small or moderate(e.g., modulation of less than 64-QAM or a Galois Field of less than64). The nonbinary regular LDPC cycle codes have a parity check matrixwith a fixed column weight of 2 and a fixed row weight. The nonbinaryregular LDPC cycle code's parity check matrix can be put into aconcatenation form of row-permuted block-diagonal matrices after row andcolumn permutations if the row weight is even, or if the row weight isodd and the regular LDPC code's associated graph contains at least onespanning subgraph that includes disjoint edges. The nonbinary irregularLDPC codes have a parity check matrix with a first portion that issubstantially similar to the parity check matrix of the regular LDPCcycle codes and a second portion that has a column weight greater thanthe column weight of the parity check matrix of the regular LDPC cyclecodes.

The encoding of the embodiments utilizing this form can be performed inparallel in linear time. Decoding of the embodiments utilizing this formenables parallel processing in sequential BP decoding, whichconsiderably increases the decoding throughput without compromisingperformance or complexity. In some embodiments, the storage requirementsfor H of cycle GF(q) codes is also reduced. Some of the exemplaryembodiments result from code design strategies, such as the codestructure design and the determination of nonzero entries of H.Extensive simulations confirm that the nonbinary regular and irregularLDPC codes of the exemplary embodiments have very good performance. Insum, this disclosure provides for the use of nonbinary regular andirregular LDPC codes in multicarrier underwater acoustic communication.The regular and irregular codes match well with the signalconstellation, have excellent performance, and can be encoded in lineartime and in parallel. Lastly, in some embodiments the use of LDPC codesreduces the peak to average power ratio in OFDM transmissions.

The apparatus, systems and methods of the present disclosure aretypically implemented with conventional processing technology. Thus,programming is typically provided for operation on a processor, suchprogramming being adapted to perform the noted operations for processingan acoustic signal in the manner disclosed herein. The processor maycommunicate with data storage and/or other processing elements, e.g.,over a network, as is well known to persons skilled in the art. Thus, inexemplary implementations of the present disclosure, programming isprovided that is adapted for a multi-carrier based underwater acoustic(UWA) signal, such that a UWA signal is sent, received and processedaccording to the disclosed apparatus, systems and methods.

Although the present disclosure has been described with reference toexemplary embodiments and implementations thereof, the disclosedapparatus, systems, and methods are not limited to such exemplaryembodiments/implementations. Rather, as will be readily apparent topersons skilled in the art from the description provided herein, thedisclosed apparatus, systems and methods are susceptible tomodifications, alterations and enhancements without departing from thespirit or scope of the present disclosure. Accordingly, the presentdisclosure expressly encompasses such modification, alterations andenhancements within the scope hereof.

What is claimed:
 1. A method for underwater acoustic (UWA)communication, the method comprising the steps of: (a) providing atleast one nonbinary, low density parity check (LDPC) code to an encoder;(b) with the encoder, encoding: (i) at least one nonbinary regular LDPCcode if the constellation size of the at least one nonbinary LDPC codeis a modulation of at least 64-QAM or a Galois Field of at least 64, or(ii) at least one nonbinary irregular LDPC code if the constellationsize of the at least one nonbinary LDPC code is a modulation of lessthan 64-QAM or a Galois Field of less than 64; (c) transmitting the atleast one encoded LDPC code through an underwater transmitter on anorthogonal frequency division multiplexed (OFDM) UWA signal; (d)receiving the at least one encoded LDPC code through an underwaterreceiver on the OFDM UWA signal; (e) storing the received at least oneencoded LDPC code; and (f) decoding the received at least one encodedLDPC code.
 2. The method of claim 1, wherein the at least one nonbinaryregular LDPC code has a parity check matrix with a fixed column weightof 2 and a fixed row weight.
 3. The method of claim 2, wherein thenonbinary regular LDPC code's parity check matrix can be put into aconcatenation form of row-permuted block-diagonal matrices after row andcolumn permutations if: (i) the row weight is even, or (ii) the rowweight is odd and the nonbinary regular LDPC code's associated graphcontains at least one spanning subgraph that includes disjoint edges. 4.The method of claim 1, wherein the at least one nonbinary irregular LDPCcode has a parity check matrix with a first portion that issubstantially similar to the parity check matrix of the at least onenonbinary regular LDPC code and a second portion that has a columnweight greater than the column weight of the parity check matrix of thenonbinary regular LDPC code.
 5. The method of claim 1, wherein the stepof encoding is performed in parallel and in linear time.
 6. The methodof claim 1, wherein the step of decoding includes parallel processing insequential belief propagation decoding.
 7. The method of claim 1,wherein the received at least one nonbinary LDPC code is stored inmemory associated with a processor.
 8. The method of claim 1, furtherincluding the step of designing the at least one nonbinary LDPC code;and wherein the step of designing the at least one nonbinary LDPC codeincludes determining the code structure design.
 9. The method of claim8, wherein the code structure design is determined based on a regulargraph, a computer search or the equivalent form of the check matrix. 10.The method of claim 8, wherein the step of designing the at least onenonbinary LDPC code includes determining the nonzero entries; andwherein nonzero entries are chosen to increase the number ofirresolvable cycles.
 11. The method of claim 1, wherein the at least onenonbinary irregular LDPC code or the at least one nonbinary regular LDPCcode reduces the peak-to-average power ratio of the OFDM signal.
 12. Anunderwater acoustic (UWA) communications system comprising: (a) anencoder adapted to receive at least one nonbinary, LDPC code and toencode: (i) at least one nonbinary regular LDPC code if theconstellation size of the at least one nonbinary LDPC code is amodulation of at least 64-QAM or a Galois Field of at least 64, or (ii)at least one nonbinary irregular LDPC code if the constellation size ofthe at least one nonbinary LDPC code is a modulation of less than 64-QAMor a Galois Field of less than 64; (b) an underwater transmitter incommunication with the encoder, the underwater transmitter adapted totransit the at least one encoded LDPC code through an orthogonalfrequency division multiplexed (OFDM) UWA signal; (c) one or moreunderwater receiving elements adapted to receive the at least oneencoded LDPC code on the OFDM UWA signal; (d) memory adapted to storethe received at least one encoded LDPC code; and (e) a decoder adaptedto decode the received at least one encoded LDPC code.
 13. The system ofclaim 12, wherein the at least one nonbinary regular LDPC code has aparity check matrix with a fixed column weight of 2 and a fixed rowweight.
 14. The system of claim 13, wherein the nonbinary regular LDPCcode's parity check matrix can be put into a concatenation form ofrow-permuted block-diagonal matrices after row and column permutationsif: (i) the row weight is even, or (ii) the row weight is odd and thenonbinary regular LDPC code's associated graph contains at least onespanning subgraph that includes disjoint edges.
 15. The system of claim14, wherein the at least one nonbinary irregular LDPC code or the atleast one nonbinary regular LDPC code reduces the peak-to-average powerratio of the OFDM signal.
 16. The system of claim 12, wherein the atleast one nonbinary irregular LDPC code has a parity check matrix with afirst portion that is substantially similar to the parity check matrixof the at least one nonbinary regular LDPC code and a second portionthat has a column weight greater than the column weight of the paritycheck matrix of the nonbinary regular LDPC code.
 17. The system of claim12, wherein the step of encoding is performed in parallel and in lineartime.
 18. The system of claim 12, wherein the step of decoding includesparallel processing in sequential belief propagation decoding.
 19. Thesystem of claim 12, further including the step of designing the at leastone nonbinary LDPC code; and wherein the step of designing the at leastone LDPC code includes determining the code structure design.
 20. Thesystem of claim 19, wherein the code structure design is determinedbased on a regular graph, a computer search or the equivalent form ofthe check matrix.
 21. The method of claim 19, wherein the step ofdesigning the at least one LDPC code includes determining the nonzeroentries; and wherein nonzero entries are chosen to increase the numberof irresolvable cycles.